1. Qb7 a) 1. ... Qf5 2. Qxe4 (and not 2. Rxe4? Qxc2+!) b) 1. ... Rxc4 2. Qh7#; c) 1. ... Qe6 2. Qh7+ c1) 2. ... Kg4 3. Rxe4+ wins the queen. c2) 2. ... Qh6 3. Qxh6+ and 4. Rxe4 or 3. Qxe4 both look good. d) 1. ... Qf3 2. Qh7+ Kg4 3. Qh3# e) 1. ... Qe2 2. Qh7+ Kg4 3. Rxe4+ wins the queen. f) 1. ... Qd1, Qd7, and Qc8 don't do anything to help g) 1. ... Kh6 2. Rxe4 or Qxe4 both look winning.
Edit: Same move as Knightstalker. Hmm.
2nd attempt:
Qb1 - the rook is threatened with capture, if it moves white either has checkmate on h6 or black's queen.
Edit: It makes for a good problem. The problem with Knightstalker's and my initial solutions is something that comes up every now and again. I've had things like that happen in games.