They can be anywhere on the Island so it can take a looooong time to find one, look high in the air, they tend to fly at about the height of a large redwood tree or half the volcano’s height. They seem to be spawn limited at around 2-4 per server so if you keep killing around say, the red obelisk and it isn’t spawning, it’s likely there are already too many wild Quetzals on the map.
Unless you have a Griffin, the easiest way to tame one is to use a Tapejara. Board the passenger seat and use whistles to chase after the Quetz while firing on it with tranqs. Be warned they do tend to fall over hazardous areas so consider bringing a cryopod and kibble to tame the Quetzal fast and store it before something attacks it.
The fun way is to jump off and grapple to your argie and whistle it to attack and since argies are slow it won’t catch up in time so tranq till it goes out also bring parachutes and consider liking so more people will know
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.
I know this is pretty common knowledge, but for the guys who always find the quetzal they're trying to tame in an area loaded with sabertooths, carnos, and rexes, all quetzals that are frightened will fly towards the coordinates 50, 50, every single time.