Mate in 2 means: white moves, such that he can mate black in his second move, regardless what black moved. Refusal chess means: you may refuse one of your opponents moves each turn. (However, when your king is in check, you must move him out of check: if you have only one possibility to do so, you lose. Note that this is slightly different from the rules of the original refusal chess: there you may not refuse the only legal move of a player.)
The first solution was send in by Ralph Betza.
White starts with 1. Qd7+ or Qe6+: one of these moves will not be refused. White refuses Ke5 (after Qd7+) or K:e6 (after Qe6+), so black must play 1. ..., Kc5. Now, white plays 2. Qc6, or when this is refused, b2-b4. Both are mate, because the black king has only one legal move to escape mate: white refuses this move and wins.