Very easy way to obtain one of these SOLO in mobile is to first build a large cage containing a revival platform (engram or tribute). Then take a fast flier with an attack/speed pteranodon following. When you find the quetzal you want, whistle the pteranodon to attack and kill the quetzal, then whistle back to neutral. Descend to the dead quetzal, retrieve its implant, and revive it at your platform for a very convenient tame. No need to fire even a single shot at it in the wild. Just be quick to retrieve its implant before something eats it.
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.