Requirements: Argentavis with about 2000 stamina with about 110 movement speed and a good amount of melee damage, Crossbow(tranq arrow)/Longneck Rifle Tranq dart), about 50 or more tranquilizers, and about 100 or more narcotics.
In order to knock out the quetzal, the best way is to find a buddy who is willing to help you out. Give the tranq gun to your friend, as he/she will be shooting. You want to get your argentavis and pick them up with the talons. Make sure you don't forget the narcotics! When you have your friend in your talons, go find the quetzal. When you find it I reccomend you land and regain stamina, but don't lose the quetzal. Once you have regained your stamina pick up your friend and proceed to knock out the quetzal. Be sure whoever is shooting to hit it in the head, or else it will take quite a while to take it down. Don't be afraid to land to regain stamina again if needed. Once the quetzal is knocked out, you need to use the narcotics and get food for him.
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
Put the honey in your hotbar (unstack them, so 1 in each slot)
Fly over him en throw the first honey on it’s body (press the hotkey from the honey and it will throw the honey) It will follow the honey that’s falling. Dino’s on the ground will eat the honey too so keep dropping honey until the quetzal is on the ground eating honey.
Place some dinosaur gates on top of him to get him stuck. Shoot it with tranq arrows and feed some prime meat
If you manage to knock a quetzal out, try starve taming. If you never tried starve taming it’s just leaving a Dino’s hunger go to half way or lower and then you start feeding. I’ve done this many times and it’s very useful taming trick.