Bring some crossbows, tranq arrows, 2 argees, a second survivor, and some narcotics for an emergency. Have you or the second player ride the argee and pick up the other, once out of stamina hop on the other argee and tranq the quetz, if out of stamina again then get on the other argee, once tamed build a quetzal taming cage on a platform saddle for it, and taming other quetzals will be much easier. Please up vote so more can see and hopefully be less clueless.
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.
To solo tame, grapple to an argentavis, and be ready to set your argentavis to passive and attack my target on command. Just maintain a process of having your argentavis chase the quetzal, when you get close enough set it passive, then fire a tranq dart/arrow. This is a long process, but it is by far the easiest way to solo tame, unless you already have a quetzal.
This tip is only if you already have a quetzal available. The easiest way to tame a quetz, is to build a quetz catcher. You can make it out of thatch and its hands down the best tactic. A lot of YouTube videos available on how to build one. Knock it out in the catcher, and you don't have to worry about it getting hit.