In chess, if a player ran out of time, how to determine if his/her opponent has the 'possibility of checkmating'?
If in doubt of being able to by player B, player A now plays both sides towards a checkmate of his choice.

In your example, player A would be able to prove it if he comes to the said checkmate somehow.

I think checkmating against King-Queen with King-Knight is impossible. The shown position (with the rook being switched to a queen) is no checkmate because the queen can just capture the knight.