For taming the quetz you should get a partner otherwise it will be very difficult to tame. Use any flying mount that you have (make sure it has enough weight to carry 2 people. Make sure to have a cross bow or tranq sniper otherwise it will take awhile to knock out. There are 2 strategies to knock the quetz out that are easy either have a person grab you on a flying mount and shoot it or having a person on a flying mount bring you very close so that you can grappling hook in to the quetz (remember to have a parachute so you don't die to fall damage when it knocks out). Then from there just shoot it till the quetz is knocked out. I would also recommend that you come prepared with at least 10 wood spike walls so you can surround your tame so it will not be harmed from the environment and lose bonus levels. Make sure to bring one crossbow with at least 1 grappling hook (bring more if you think you will miss), tranq sniper (optional), narcotics (800+), wood spike walls, and a flying mount.
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.