If you have a Quetzal, put a simple cage on it (This cage worked for me https://youtu.be/oHx1KeJX408) and have it follow your Argy, check every once in a while it doesn't get too far. Land your Argy on Quetz to regain stamina. Quetz will never run out of Stamina.
When you find your wild Quetz get in your Quetz and have Argy follow you and not your Quetz in case you fall. Always carry lots of parachutes in case you fall.
Get wild Quetz stuck in cage and take it down. When your Quetz gets low in stamina, jump on your Argy and let it regain to full. Repeat process until done. You can also check stamina of your Quetz will flying your Argy and getting close to your Quetz. No meat and stimz required.
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.