After you manage to find one of these (usually high in the air, above quetzals. I've had the most luck around 50, 50) you'll want to make a trap. A Trapejara. Make a 3x3 cage out of doorways and put a ceiling on it.
Find your victim.
Fly level to it, using the back center doorway to line it up. Now for the chase, you'll want a quetzal with... a lot of movement speed. Mine is 211 and I struggle sometimes. Fly straight for the Tapejara, and once it clips into the structure, GET OFF OF THE QUETZAL. Your new buddy is trapped and afraid.
If you fail, let it fly out of render distance and try again. It won't be aggro'd anymore and it'll be flying normally do you have another shot.
If you just want to keep chasing it, wait about 15 seconds and it'll re-aggro, pausing for a second to do an animation. This is when I usually catch them.