If you already have a quetzal build a trap on it with a trapdoor on the top. Fly infront of the wild quetzal and make it fly inside. Unmount at a slight angle and the wild quetzal will get stuck. Now you can stand there mid flight and tame it.

Notice:

1 dont move your quetzal or the wild one could slip out, even when asleep.

2 add gates and a ramp to lure in ground animals.

3 gate to the front to trap smaller birds (just fly faster than them and "net" them like a jellyfish.

4 add ceilings around the trap as a platform to move on.

5 build the trapdoor and front gate in a way so they are open while you are flying (auto open and close).

6 add a ramp or path so you can get to your quetzal and mount it while flying.

Personally i added a fridge and bed to store kibble and let me spawn there.