Would you like to read a short paragraph that explains how to win with almost any position in a queen-versus-rook end game? That would take a book, not a short paragraph. The following summarizes a few of the key patterns to look for in winning the Q-vs-R pawn-less end game, yet the list is incomplete.
Diagram-1 (Black to move)
Probably the best-known winning position in this kind of end game is the Philidor. We assume it’s black’s move, for it’s a zugzwang (and with white to move, the attacker can transfer the move back to black).
The above position, a wishbone, also has the defending king on the edge, but here it’s far from a corner. Probably white’s best move here is Qb1+, following by Kc3, a procedure that Derek Grimmell calls “turning the thorn on its side.”
Third Rank Defense
Diagram-3 (White to move)
Of all the queen-versus-rook defenses in which the defending king is on the edge of the board, this one is probably the most difficult for the attacker to break down. There is a formula for winning, but it’s precise and it’s non-intuitive.
Diagram-4 (Black to move)
The Trapezoid may have been named by Derek Grimmell, whose youtube videos on Q-vs-R end games are highly enlightening. Black to move has nothing that does not result in a quick checkmate or quick loss of the rook with no stalemate possibility. With white to move, I’d recommend Qb2, with few options for black.
In the queen-versus-rook Philidor, the defending king has only one legal move, and it results in the queen pinning the rook and capturing it on the next move.
Youtube chess video by Derek Grimmell