We have found that the best way to tame a quetz is to trap it with another quetz. On a quetz platform saddle, it must be 3 walls high, with some ceilings to keep it in, and some open spaces on the sides, but walls to keep it in as well. Leave the back open and fly in front of the quetz. Let the quetz fly into the trap(the front-middle of the platform must be open so the head can go through far enough). Once the quetz is in, DO NOT MOVE, or it will go right through the trap. Put walls in the middle of the back of the catcher to trap it in, and tranq it. Once it's down, you can tame it in the air, or just move a little and it'll drop out to the ground and you can put spikes around it to tame it where it fell.
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.
Easy way : Bring a harpoon gun with net ammo and BASICALLY Shot the quetzal and make sure it falls in a safe area and then trap it as fast as you can and Tranq it with your ammo of choice and Normal Gates can work and even if it wakes up it shouldn’t be able to escape easily and to net it again you have to De-agro it