On the center map, when the Quetzal is hit enough times it will ALWAYS eventually fly in-between the two mountains at coordinates 30 55 where you can set up a chain bola catcher thing or something. With an argy, hold a diplocaulus of something that isn't very strong to hit the Quetzal to this area so it doesn't die. Good luck.
So here is a story once on mobile I found one I am playing solo so I go and gear up for the tame it takes me from the swamp to the winter biome to the redwood to the north beach to the south beach back to the swamp I finally bring it down over the swamp instantly it gets attacked by a spino I am on a pterronadon I try to lead it away and I lost both my beloved pterronadon and the quezt so my tip be careful where you bring one down
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
In order to solo tame a quetz, you could use an argentavis and a pulmiscorpius. Just pick up the pulmiscorpius, and do whistle attack target while looking at the quetz, your pulmiscorpius should start attacking the quetz, raising it's torpidity until it is knocked out.