Best quetzal cage I've found is made of gates. Just need a 3x3 base, four normal gates in a square (leave small gaps between them, and optionally a pillar with a ceiling. Just drive until the quetzal is between the gates and stop/hop off yours - it will be unable to flee if its body is inside the gates and you're free to knock it out or kill it if it doesn't meet your standards.
Farming them near red obelisk is ideal for me, if you don't like the level of the one you trapped you can kill it and another will spawn almost immediately.
The fun way is to jump off and grapple to your argie and whistle it to attack and since argies are slow it won’t catch up in time so tranq till it goes out also bring parachutes and consider liking so more people will know
If you drop a quetzal due to knockout, and it lands next to a giga...just know, the quetzal belong to the giga. Now you get to watch your would be quetzal get eaten by a giga. Now you can go home and cry.