White cannot checkmate Black with the White Queen alone. White King must trek across the board and assist in the execution. Checkmate with a King and Queen can be accomplished only at the board edge.
From diagram, White proceed with the endgame as follows:
1.Qb5
With this move, White is trapping the Black King in a "box". The box method of checkmate is easiest, though slightly faster methods are also possible. The strategy here is to make the box smaller and smaller. After White Queen has trapped the Black King on the back rank, White King will march over and assist the Queen.
1...Ke6 2.Qc5
1...Kc7 2.Qa6
White avoid playing 2.Qc6+, because a response of 2...Ke5 from Black would allow the Black King more room.
2...Kf6 3.Qd5
2...Kd7 3.Qb6
White does not lose track of his goals by uselessly checking his opponent.
3...Kg6 4.Qe5
4...Kf7 5.Qd6
4...Kh6 5.Qg3, thus trapping the Black King on the edge of the board.
Now Black is trapped on the last two ranks.
5...Kg7 6.Qe6
6...Kh7 7.Qg4
6...Kf8 7.Qd7
Black King is trapped on the side of the board.
7...Kh6 8.Kb2 Kh7 9.Kc3 Kh8
10.Kd4 Kh7 11.Ke5 Kh8 12.Kf6 Kh7 13.Qg7#
10.Qg6?? Stalemate
Check my post on how to check mate with lone rook here.
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