Don't wait until you're level 85 and is able to tame a griffin. Even though it is the easiest method to tame a quetz for the first time, its not worth the wait since Tapejaras exist. Make sure to fly high enough so that your tapejara don't automaticlly fly to the ground and then swap to the passanger seat. Whistle target while aiming at the quetz and your Tapejara will fly towards the quetz. If you're near enough, whistle passive and shoot at the quetz. Repeat this and you will have a knocked quetz. And don't worry, Tapejaras stamina won't slowly dissapear when you're on the passanger seat. And don't forget to bring a parachute just in case
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.