When looking at the position, the first thing I notice are of course the obvious forcing looking moves. If I move the bishop the queen is attacked by a rook. That is duh of course...but because of that I start to look for ways the bishop can move with some threat besides just the queen.
I also chunk together various common tactical motifs. I see that if my queen reaches d5...AND if I get the enemy queen to not defend f7, then I will win a rook or smothered mate. That whole tactical sequence is something I see a lot so its almost like one move to me. So I see that moving my bishop will not only threaten the queen, but the queen may be fairly limited in its moves.
Another tactical pattern chunk that I see...because I am interested in moving the knight because I am interested in the white queen having access to d5, is Nb5-nxc7 Rb8 Bxa7, trapping the rook. Again that sequence is very common so it is like one move to me.
Something that makes finding the tactics here rather difficult is that there are so many different tempting ideas...for example the move f5 looks tempting because I can see a lot of potential action on the f file, especially coordinated with Qd5+.
I played the first move in the puzzle but I was really kinda guessing. I wasn't entirely sure it was better than Nb5...and I still just felt like playing f5...but I could also see that black has a lot of 'things' going on himself and therefore could probably repair his position if I don't do something very forcing...nb5 and f5 just did not feel like it came with quite enough of a threat. But I did not see that Qf6 was forced. In fact, I still don't. Qd7 Ne6 re8 looks unpleasant, but I don't see the win.
But anyway after Qf6 it was easy to play Nb5 because it was something 'in the air' from the start and c7 is unprotected. But again...I don't really see why qxf4 is the 'only' move.
I did not guess 19.qxq. I wanted to play 19 Qd5+ kh8 20 Bg3 with the idea of nf7+...and I still don't see why that's wrong.
I once worked on a tactics puzzle from GM Aagaard. I stared and stared at the position but nothing came. I finally gave up and read the solution. It was a couple of forced moves that climaxed with White Qd5 forking Black's castled king on g8 and his queen rook on a8 which was loose because the QB was still on c8 like your position.
Aagaard commented that any master would notice the possibility in that configuration. Well, I'm not a master and I hadn't noticed, but I do now, although not consciously. I get a tingle when I see the diagonals to the king and rook are open like that, just like I do when I see a king open to a back row mate.
But that comes from experience, not a set of conscious rules for move selection.