Get on a Tapejaras front seat as soon as you find a quetzal and make it attack the quetz. The quetz is a bit faster, but he will fly in curves so the Tapejara can easely follow it. Just tranq it with your crossbow/rifle as you can shoot from the front seat!
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
Just shoot it with a dodo arrow (unlocks at level 42) and it should agro you, you don't even need a Tapejara or a griffin for this, all you need is a Pteronodon, parachute and a trap (The trap is optional) and just fly up to it, jump off your Pteronodon and activate the parachute then shoot it with the dodo arrow. While it is agro just keep shooting it with either a crossbow with toxicant/tranq arrows or a long neck rifle with tranq darts. Just make sure you don't go near water because the Quetz might fall in and drown when it gets knocked out. Happy Quetz taming
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.
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.