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?
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).