user16801: If my opponent brings out his/her queen early in the game, I usually attack it with my own queen. The opponent can either accept the trade or reject it. If he/she accepts, I have no questions at all. The game continues, and I am happy.
The question is: what if the opponent refuses to make any trades after I suggest the trade of queens? E. g. I am playing white; my opponent, black. He brings out the queen, I attack it. Then he simply moves the queen to a safe square. Quite 
often it results in me being checkmated in about 10 next moves, or I lose a piece.
So if the opponent constantly refuses to trade queens in the opening, what should I do? Keep attacking the queen (possibly searching for forks/pins/skewers?), or develop my minor pieces, or try to defend my king? Thanks.

user-1: Here is a demonstrative game to show how you should deal with a 'queen raid'.
1. e4 e5 2. Qh5 Nc6 3. Bc4 g6 4. Qf3 Nf6 5. Qb3 Nd4 6. Bxf7+ Ke7 7. Qc4 b5 8. Qc3 Kxf7 9. h3 Bb4 10. Qg3 

Gain tempo by attacking the queen, and attempt to launch a quick attack. c2 and f2 are spots to target.

user598: Ideally, you should be setting up trades that are good for you and bad for your opponent - equal material but better positioning for you, for example. At the same time, you need to make your moves such that if your opponent is declining the trade, it still leaves you with a better position. 
If you spend a move attacking a queen, and the queen "retreats" but actually puts itself in a better position, then perhaps don't make that attack yet. All you're doing is throwing away your tempo.

user17800: It doesn't really mean anything to attack (or exchange) a piece if you don't get anything from it. Also, it can end up getting you in a worse position if you don't pay close attention.
If he brings his queen out, unless there is a threat, keep bringing your pieces out and develop your game. And if you can attack his queen while developing a piece to force him to move it, it'll be much more beneficial for you than trading queens at the beginning.

user15221: You seem to be saying "I'm a weaker player than my opponent, but if I can trade off all my material, then obviously that'll be a draw by insufficient material. How do I force my opponent to accept this draw, instead of going for a win?" And there isn't much answer to that other than "Be a better chess player". If your opponent is willing to move their queen back to avoid a trade, then apart from tempo, you're in the same position as if they hadn't moved the queen out at all, so you're pretty much asking how to trade queens in general. And there isn't really any way to do that other than getting your opponent in a position where they will lose material and/or position if they don't trade, and asking "How do I create threats to force a trade" is not only an extremely broad question, it's not a good question. You shouldn't be trying to create a situation where your opponent can avoid material/position loss by trading their queen, you should be creating situations where you opponent can't avoid material/position loss no matter what they do.
Another thing to consider is that while it's sometimes easier to get your queen in a position where it can trade for the opposing queen than to get another piece in a position where it can attack the queen, that's not usually the case, and once a piece is threatening their queen, having that piece be your queen does not make it any more of a forced trade. If anything, attacking with your queen rather than another piece gives your opponent more options. All else being equal, any situation where your queen is attacking theirs would be improved by it being another piece attacking. The only advantage to it being your queen that's attacking is baiting them into a blunder, and relying on your opponent to make a mistake isn't good strategy.

user2789: Trying to encourage a queen trade in this sort of situation is usually the wrong strategy.
The exaggerated version of the answer is this.  Your opponent has just proved that they don't know how to use their queen properly.  Instead of saying, "That's OK. I'll trade off the queens so this isn't a problem for you", you should say "I'll keep exploiting this big weakness in your play."  This isn't completely true, because it's not always wrong to bring your queen out early. But it's a good rule of thumb.
Bringing out the queen early is (usually) bad because the queen is valuable so, every time it's attacked, it must be moved out of the way. Unless it's attacked by the queen, it must be moved: defending it doesn't work, because the attacker is less valuable. So, when your opponent brings out their queen early, you can usually make sensible developing moves that attack it, which forces your opponent to keep moving their queen around. If you're lucky, they'll get their queen trapped and you'll win it.  Otherwise, you're still in a great position because you've developed most of your pieces and probably got your king to safety, while your opponent has just been moving their queen around.

user8472: There is no need for an early queen trade. Just develop your pieces, if possible by attacking the queen. Here is an example from the Wayward Queen Attack also known as Patzer Opening.
[fen ""]

1. e4 e5 2. Qh5 Nc6 3. Bc4 g6 4. Qf3 Nf6 5. Ne2 Bg7 6. Nbc3 O-O 7. d3 d6


2.Qh5 threatens to take e5, so we defend it by developing a piece: Nc6 (d6 is also possible).  
3.Bc4 threatens to mate on f7, so we defend it by attacking the queen with g6, creating a square for our bishop as a side effect.
4.Qf3 renews the threat, so we develop another piece, Nf6 and block the queen's access to f7.
Then we develop our bishop, castle and get a nice game with a development advantage.

The best punishment for an early queen development is not a queen trade, but using this queen to develop your minor pieces with a tempo: by forcing the queen to move again and again.
Update
In the example you gave in the comment, you play just the same way: Develop your other knight, get castling in, and have a good game. There is a reason why you don't see this line in games by players with a decent rating.
[fen ""]

1. e4 c6 2. Nf3 Qa5 3. Nc3 d5 4. Be2 dxe4 5. Nxe4 Bf5 6. Ng3 Bg6 7. O-O Nd7 8. d4

First let me say that 2.Nf3 is not the best move against the Caro-Kann. d4, Nc3 or even c4 are better and therefore more common. See my Caro-Kann study for more ideas.
After 2…Qa5 you just play Nc3 which develops a piece and unpins the d-pawn. Then you clear the f1 square by developing yet another piece, Be2, castle as soon as possible, and then play d4.
Black is at this point seriously behind in development: the entire kingside is still in the default position while you have already some very annoying moves like Nh4,  c4, or Bf4. Your queen on d1 is keeping everything together. It's not in the way of any other piece and much better than Black's queen – and you didn't even move it!
So, again: stay calm, develop your pieces, castle, and then question your opponent's position with your own minor pieces.

16801: <p>If my opponent brings out his/her queen early in the game, I usually attack it with my own queen. The opponent can either accept the trade or reject it. If he/she accepts, I have no questions at all. The game continues, and I am happy.</p>

<p>The question is: what if the opponent refuses to make any trades after I suggest the trade of queens? E. g. I am playing white; my opponent, black. He brings out the queen, I attack it. Then he simply moves the queen to a safe square. Quite 
often it results in me being checkmated in about 10 next moves, or I lose a piece.</p>

<p>So if the opponent constantly refuses to trade queens in the opening, what should I do? Keep attacking the queen (possibly searching for forks/pins/skewers?), or develop my minor pieces, or try to defend my king? Thanks.</p>
 -1: <p>Here is a demonstrative game to show how you should deal with a 'queen raid'.</p>
<pre><code>1. e4 e5 2. Qh5 Nc6 3. Bc4 g6 4. Qf3 Nf6 5. Qb3 Nd4 6. Bxf7+ Ke7 7. Qc4 b5 8. Qc3 Kxf7 9. h3 Bb4 10. Qg3 
</code></pre>
<p>Gain tempo by attacking the queen, and attempt to launch a quick attack. c2 and f2 are spots to target.</p>
 598: <p>Ideally, you should be setting up trades that are good for you and bad for your opponent - equal material but better positioning for you, for example. At the same time, you need to make your moves such that if your opponent is declining the trade, it still leaves you with a better position. </p>

<p>If you spend a move attacking a queen, and the queen "retreats" but actually puts itself in a better position, then perhaps don't make that attack yet. All you're doing is throwing away your tempo.</p>
 17800: <p>It doesn't really mean anything to attack (or exchange) a piece if you don't get anything from it. Also, it can end up getting you in a worse position if you don't pay close attention.</p>

<p>If he brings his queen out, unless there is a threat, keep bringing your pieces out and develop your game. And if you can attack his queen while developing a piece to force him to move it, it'll be much more beneficial for you than trading queens at the beginning.</p>
 15221: <p>You seem to be saying "I'm a weaker player than my opponent, but if I can trade off all my material, then obviously that'll be a draw by insufficient material. How do I force my opponent to accept this draw, instead of going for a win?" And there isn't much answer to that other than "Be a better chess player". If your opponent is willing to move their queen back to avoid a trade, then apart from tempo, you're in the same position as if they hadn't moved the queen out at all, so you're pretty much asking how to trade queens in general. And there isn't really any way to do that other than getting your opponent in a position where they will lose material and/or position if they don't trade, and asking "How do I create threats to force a trade" is not only an extremely broad question, it's not a <em>good</em> question. You shouldn't be trying to create a situation where your opponent can avoid material/position loss by trading their queen, you should be creating situations where you opponent <em>can't</em> avoid material/position loss no matter what they do.</p>

<p>Another thing to consider is that while it's sometimes easier to get your queen in a position where it can trade for the opposing queen than to get another piece in a position where it can attack the queen, that's not usually the case, and once a piece is threatening their queen, having that piece be your queen does not make it any more of a forced trade. If anything, attacking with your queen rather than another piece gives your opponent more options. All else being equal, any situation where your queen is attacking theirs would be improved by it being another piece attacking. The only advantage to it being your queen that's attacking is baiting them into a blunder, and relying on your opponent to make a mistake isn't good strategy.</p>
 2789: <p>Trying to encourage a queen trade in this sort of situation is usually the wrong strategy.</p>

<p>The exaggerated version of the answer is this.  Your opponent has just proved that they don't know how to use their queen properly.  Instead of saying, "That's OK. I'll trade off the queens so this isn't a problem for you", you should say "I'll keep exploiting this big weakness in your play."  This isn't completely true, because it's not always wrong to bring your queen out early. But it's a good rule of thumb.</p>

<p>Bringing out the queen early is (usually) bad because the queen is valuable so, every time it's attacked, it must be moved out of the way. Unless it's attacked by the queen, it must be moved: defending it doesn't work, because the attacker is less valuable. So, when your opponent brings out their queen early, you can usually make sensible developing moves that attack it, which forces your opponent to keep moving their queen around. If you're lucky, they'll get their queen trapped and you'll win it.  Otherwise, you're still in a great position because you've developed most of your pieces and probably got your king to safety, while your opponent has just been moving their queen around.</p>
 8472: <p>There is no need for an early queen trade. Just develop your pieces, if possible by attacking the queen. Here is an example from the <em>Wayward Queen Attack</em> also known as <em>Patzer Opening</em>.</p>

<pre><code>[fen ""]

1. e4 e5 2. Qh5 Nc6 3. Bc4 g6 4. Qf3 Nf6 5. Ne2 Bg7 6. Nbc3 O-O 7. d3 d6
</code></pre>

<ul>
<li><code>2.Qh5</code> threatens to take <code>e5</code>, so we defend it by developing a piece: <code>Nc6</code> (<code>d6</code> is also possible).  </li>
<li><code>3.Bc4</code> threatens to mate on <code>f7</code>, so we defend it by attacking the queen with <code>g6</code>, creating a square for our bishop as a side effect.</li>
<li><code>4.Qf3</code> renews the threat, so we develop another piece, <code>Nf6</code> and block the queen's access to <code>f7</code>.</li>
<li>Then we develop our bishop, castle and get a nice game with a development advantage.</li>
</ul>

<p>The best punishment for an early queen development is not a queen trade, but using this queen to develop your minor pieces with a tempo: by forcing the queen to move again and again.</p>

<p><strong>Update</strong></p>

<p>In the example you gave in the comment, you play just the same way: Develop your other knight, get castling in, and have a good game. There is a reason why you don't see this line in games by players with a decent rating.</p>

<pre><code>[fen ""]

1. e4 c6 2. Nf3 Qa5 3. Nc3 d5 4. Be2 dxe4 5. Nxe4 Bf5 6. Ng3 Bg6 7. O-O Nd7 8. d4
</code></pre>

<p>First let me say that <code>2.Nf3</code> is not the best move against the Caro-Kann. <code>d4</code>, <code>Nc3</code> or even <code>c4</code> are better and therefore more common. See my <a href="https://lichess.org/study/4Aqycynk" rel="noreferrer">Caro-Kann study</a> for more ideas.</p>

<p>After <code>2…Qa5</code> you just play <code>Nc3</code> which develops a piece and unpins the d-pawn. Then you clear the <code>f1</code> square by developing yet another piece, <code>Be2</code>, castle as soon as possible, and <em>then</em> play <code>d4</code>.<br>
Black is at this point seriously behind in development: the entire kingside is still in the default position while you have already some very annoying moves like <code>Nh4</code>,  <code>c4</code>, or <code>Bf4</code>. Your queen on d1 is keeping everything together. It's not in the way of any other piece and much better than Black's queen – and you didn't even move it!</p>

<p>So, again: stay calm, develop your pieces, castle, and then question your opponent's position with your own minor pieces.</p>
