If you already have a quetz with a platform, build a trap on the platform. 2 thatch foundation with 1 thatch ceiling around each edge (should have a 3x3 tile floor), then build thatch door frames and build them 2 high on every other tile on each edge except for the back. Have 4 thatch doorframes and at least 6 thatch ceilings ready. You will fly in front of the Wild quetz and let it fly into the box, once it's in the box, dismount and place the doorframes on the rear edge behind the quetz, then place the ceilings on all your doorframes and in the centre on the ceiling square. You can now knock it out and it cannot escape.
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 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.