At Friday’s club meeting, we showed a video on achieving checkmate with only a King and Queen. The Queen is used to drive the opponent’s King into a corner by using the Knight’s shadow technique. Care must be taken to leave some room for the opponent’s King to move, to avoid a stalemate position. The King then drives up the board to support the queen in the final checkmate move.
There is also a good article on Wikipedia that describes checkmate with only a King and Queen.