Does a KQQ vs KQ endgame position with perpetual check exist? 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 &quot;5k2/8/8/8/8/5q2/7Q/6QK w - - 0 1&quot;]

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).