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.
If you manage to knock a quetzal out, try starve taming. If you never tried starve taming it’s just leaving a Dino’s hunger go to half way or lower and then you start feeding. I’ve done this many times and it’s very useful taming trick. Upvote if you agree.