Just tamed lvl 196 from wyvern. Needs some patience as you land one hit at time (you can try 2 longnecks) , but it has one advantage, you can sprint after hit infront of quetz and change his direction as it tries to escape from you. Another option is to use low level slingshot to make him flee, for example from water to land. Either way you just fly infront and dismount wyvern to take a shot, you can even walk up his head, it's impossible to fall down. Don't forget to whistle wyvern on follow to stay in air.

ON XBOX/PS/PC:

WITHOUT ANOTHER QUETZAL:

Use a griffin or a tapejara to tame it. With a griffin, control it’s direction of movement and knock it out with tranq darts. With a tapejara, whistle attack target and whistle passive using the corresponding buttons.

WITH QUETZAL:

Build a large stone WALL container and fly in front of it head on. Make sure it gets trapped first, then close it off with walls. Knock it out.

ON MOBILE:

Use dodo arrows to shoot it and lead it into behemoth gates. Make large walls and ceilings to block it in. YOU CANNOT USE ANOTHER QUETZAL, the wild one will just glitch through it.

If you need to grab a second quetzal for breeding and you already have one, I recommend using your first quetzal to chain bola the secons one and place stone gates on its neck, body, and legs. You can then proceed to tranq it no problem. Oh, and you might want to bring some spiked walls to keep the predators out.

For taking a quest first time, first game a low lvl so you can later catch a high 100+ quetz. First get a high level argentavis and get flak armor with water and food. Get 10 grapples and 10 parachutes, preload the crossbow with the grapple and add it to your inventory, also put the parachutes in your bar. If it’s a level 15 or so get 30-40 tranq darts or get 20 shocking tranq darts. Find the quetz and get near it in the air have your crossbow in your hand before you get on the argen, when you are near the quetz jump off your argen and equip the para, then immediately grapple argen. When you’ve done that, shoot a dart at the quetz then send your argen after it using the whistle, since quetz fly towards 50,50 you are near water, count how many times you hit it to make sure it doesn’t knock out over water, if it knocks out near 50,50 make sure you immediately put spike walls all the way around it. Once you’ve done that kill all creatures around, then give it prime and 200 narcotics. Cryo

If you want to tame one set a tamed Tapejara to follow you but you have to be off your tame first then find change seats a quetzal set it to attack the quetzal and shoot some tranqs and you can set your Tapejara to follow you when you are of it and when you fly it it will stay in the air so it is a easy way to kill alpha’s.

Best solo method that I've found, by far, is to use a Archaeopteryx with a Tapejara. Once you find a Quetz, jump off and switch to front passenger. Whistle attack while shooting tranqs, then whistle passive once you get too close. I bring Argies and set target and follow distance both to high, make them follow the Tapejara. When you eventually knock the Quetz, whistle the Argies to follow you to defend it wherever it lands. Make sure all the Argies are set to "Only attack conscious".

Any taming food or kibble I bring, I place in the Tapejara and have them set to passive when I land. Once you knock a Quetz out of the sky, you can use crouch to go straight down with the Tapejara so you don't lose where the Quetz lands.