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.
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.
Quetzals, one of the best dinos. Quetzals can easily be tamed with a buddy, all you need is a flying dino that can pick up people or has a multi-seat saddle. Load up with the appropriate tranq arrows, narcotics, food and set off across the map to find one. Low levels are brilliant for beginners as they can serve as brilliant carriers due to their high base weight stat. Higher levels however mainly are used as "Battle Quetzals", alongside the high stats it will receive from being tamed, pumping health and melee damage is the way to go. With that being said you'll be able to tank base defences such as PSX and Auto Turrets, aswell as kill dangerous dinos with ease. Maybe either of those don't tickle your fancy, then hey, make a "Party Quetzal"! Strap on some chairs, tables, food and lots more onto the Quetzal's Platform Saddle and hover in the air as you dance to Darude Sandstorm being blasted through someone's mic in game chat. If you can get a quetzal, do consider as they are brilliant!
When you tame your first quetzal, you should make it into a quetz trapper... if you are wanting more... if you don't know what a quetz trap is, go to YouTube! If you can tame a low level one to start! Being able to tame as many quetzals as you want with the trap, is RIDICULOUS!
The Quetzal is a behemoth of a creature with base stats that top the charts (health, stamina, weight, and torpor). The Quetzal's platform saddle and ability to regain stamina when paused in mid-air (dismounted) make it an excellent choice for a mobile air base or even a caravan for transporting and gathering metal and other supplies on harder to reach areas. While its melee strength isn't as high as some might expect, its health and ability to carry weapons make it suitable to use as a war bird. Its preference for Rex Kibble make it difficult to tame, and even more so difficult to knock out in the air.