King + Queen v King? The method is to push the king back to an edge of the board by ensuring that it can only move in that direction by advancing your king and queen in lock-step.
The slow way to do it is use the queen as a rook, as in K+R v K.
The efficient way to do it is to advance the queen into one of the squares protected by the king to minimise sideways movement. Just be careful that when the king isn't in check and is at the edge of the board, that it has a legal move or you'll get a stalemate.
Does this cover it sufficiently or do you need more detail?
[caveat: I suspect that I haven't understood the question.]
I'm interested in mastering this (pawnless) endgame from the Philidor position and beyond. I'm particularly interested in any coach who has a 'system' or 'method' for deciding what moves to choose in specific situations rather than someone who just 'plays it by feel', but any help is appreciated.