[Corrected: most of the alternative solutions I suggested fail to g4+]
There must be some mistake in the diagram, because R(either)xR+
is a non-quiet move that forces mate in 2. Quiet moves don't work
because of the defense Rc8+ or Rxh8. Except that the FEN indicates
that the wP is on g2, not b7 as it seems, and that makes Qa8! a quiet
solution (Zugzwang: if RxR(either) then QxR is mate; other Rook moves
are captured by a Rook for mate; and the pawn is still pinned).
To make this unique the Queen can be put elsewhere on the diagonal,
[fen "7r/8/8/3q1k2/8/8/6PR/r5RK b - - 0 1"]
I see that Dag Oskar Madsen already suggested this fix in a comment.
Possibly of interest: while trying to Google this position I ran across
this mate-in-2 problem,
credited to T.A.L.Herlin 1852, which might have been the inspiration
for the present problem though its solution is not as nice or as quiet:
[fen "kr5R/rp6/8/8/4Q1K1/8/8/8 w - - 0 1"]
1 Qe8! (threat 2 QxR#) Rxe8/Ra4+ 2 Rxe8/Qxa4#!