Celtic can complete the feat against Aberdeen on Saturday in the Scottish Cup final.
"We may seem calm, but we always have pressure, when we walk on that pitch and see the crowd," Boyata said.
"There is always pressure in our body. But when we start playing we don't think about what's going on around us."
Celtic broke several records on their way to finishing the league season unbeaten, and defeated Aberdeen in the League Cup final last November.
Boyata says that the players will take encouragement from having seen off Saturday's opponents in all five previous meetings this season, but will not take victory for granted.
"The treble is a big word. Invincible was another big word," Boyata said. "The treble is another thing that we're trying to achieve. It will be hard, we will feel pressure.
"We need 100% focus, we know it's going to be a big achievement if we manage to win this game and we all just want to finish it.
"It will be very sad and hard if we don't finish on a high. For anybody who's been in the position where everything is well and then the last second you lose a game, you don't feel good.
"The cup is very important, but it is for Aberdeen as well so it will be a hard game. We're going to be ready to give everything because it's the last game as well.
"We know we won't have any more chances after that, so there's no time for mistakes. It's the last game of the season."
Dedryck Boyata says the Celtic players will cope with the pressure of trying to earn only the fourth treble in the club's history.