user-1: In such an endgame situation (no specific position), where White isn't pinning the Black queen on the first move, is it possible for Black to get a perpetual check if they move first?

user307: Yes, it is possible. Here is an example:
[Variant "From Position"]
[FEN "5k2/1Q6/8/8/7q/8/8/6QK w - - 0 1"]

1. Qh2 Qe1+ 2. Kg2 Qe2+ 3. Kg3 Qe5+ 4. Kh3 Qh5+
5. Kg2 Qe2+


user7297: The position given by Akavall is indeed a draw by perpetual, but it's a bit difficult to see because White has many different options at some moves. They can even choose to sacrifice the h2 queen, which effectively ends the perpetual (but it's still a draw).
A position where this is much easier to see is the following one:
[FEN "5k2/8/8/8/8/5q2/7Q/6QK w - - 0 1"]

1. Qgg2 Qd1+ 2. Qhg1 Qh5+ 3. Q2h2 Qf3+

White has two options each move, and the black queen simply moves to the opened line/diagonal. As mentioned in the comments, White could sacrifice a queen on f1 or h3, but even then Black can force a perpetual on f1 and h3 (though the easiest way to draw is then trading the last pair of queens).

-1: <p>In such an endgame situation (no specific position), where White isn't pinning the Black queen on the first move, is it possible for Black to get a perpetual check if they move first?</p>
 307: <p>Yes, it is possible. Here is an example:</p>
<pre><code>[Variant &quot;From Position&quot;]
[FEN &quot;5k2/1Q6/8/8/7q/8/8/6QK w - - 0 1&quot;]

1. Qh2 Qe1+ 2. Kg2 Qe2+ 3. Kg3 Qe5+ 4. Kh3 Qh5+
5. Kg2 Qe2+
</code></pre>
 7297: <p>The position given by Akavall is indeed a <a href="https://syzygy-tables.info/?fen=5k2/1Q6/8/8/7q/8/7Q/7K_b_-_-_0_1" rel="noreferrer">draw</a> by perpetual, but it's a bit difficult to see because White has many different options at some moves. They can even choose to sacrifice the h2 queen, which effectively ends the perpetual (but it's still a draw).</p>
<p>A position where this is much easier to see is the <a href="https://syzygy-tables.info/?fen=5k2/8/8/8/8/5q2/7Q/6QK_w_-_-_0_1" rel="noreferrer">following one</a>:</p>
<pre><code>[FEN &quot;5k2/8/8/8/8/5q2/7Q/6QK w - - 0 1&quot;]

1. Qgg2 Qd1+ 2. Qhg1 Qh5+ 3. Q2h2 Qf3+
</code></pre>
<p>White has two options each move, and the black queen simply moves to the opened line/diagonal. As mentioned in the comments, White could sacrifice a queen on f1 or h3, but even then Black can force a perpetual on f1 and h3 (though the easiest way to draw is then trading the last pair of queens).</p>
