Equip your crossbow with a grappling hook. Fly your Argy around and look for a quetzal. After finding, jump off your Argy, and grappling hook it. Whistle attack target on the Quetzalcoatlus and start shooting tranq dart/arrows at the Quetzalcoatlus. Note: Make sure to whistle passive right before the Argy reaches the Quetzalcoatlus so the Quetzalcoatlus doesn’t die.
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.