Besides using a tapejara, you could use a quetz catcher. You can google this for photos. Basically, if you have a quetz, or someone you know does, you can build a catcher on the back (thatch is fine). After "catching" (making it fly into the catcher) the quetz, jump off the quetz onto the platform saddle and add some walls in the open spaces (where it flew into). Then, you can shoot it with darts or arrows till it's down. Or, you can have a friend shoot at it from another bird/platform.
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.