On the center map, when the Quetzal is hit enough times it will ALWAYS eventually fly in-between the two mountains at coordinates 30 55 where you can set up a chain bola catcher thing or something. With an argy, hold a diplocaulus of something that isn't very strong to hit the Quetzal to this area so it doesn't die. Good luck.
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
Use a netgun when it flies above ground. Once netted, either have a trap ready for high levels or shoot tranq arrows into their eyes(more dmg). It you have a buddy, have them shoot with tranq darts while you shoot into the eyes with arrows. Be careful though as they’re kinda wimpy at times. Super hard to tame while solo without a net gun.
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.
You can net a quetzal but before you do clear the area of any hazards and hop on you creature and bet it (Make sure you don’t shoot it when it is asleep because when is has the bet on it looks like it’s asleep)
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.