11

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?

32

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).

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13

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+
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