If you don't own a quetzal or a tapejara already. This will be tricky. I suggest asking for some help on your server. However, if you do own a quetzal and want another one, here's some tips. You need to build a trap on your current quetzal's platform saddle. It needs to be 3 high and 3 long. You can use thatch. Make it 3 wide/long using thatch foundations. Make it 3 high using thatch door frames. Then add thatch ceilings. Leave where you're siting open. Don't cover that with door frames just yet. When you're flying at the quetzal make sure you're going at it head first. Once it's flying in place in your trap stand up and put the thatch door frames behind where you're sitting, connecting it to the foundations to the ceilings. Then start tranqing wait every 10 seconds while shooting it so you don't kill it. 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
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.