If you already have a quetzal and want to tame another one, make a 3x3x3 structure on the back of the quetz platform you already have and leave one side open for the quetz to fly in. then fly in front of the quetz you are looking to take, and right before it fly's out of the front of your cage, hop off the quetz and it will become solid. Once it is inside, place a wall behind it on the wall you left open. It will be stuck, and you are free to tame it with no chance of anything hostile ruining your tame. Since you aren't riding the quetz, it will regen stamina in the air and you can hover until the tame is done. Once its done, just destroy the wall you put to trap it and fly straight and it will fall out and follow you. Enjoy your quetz :)
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.