If you already have a quetzal build a trap on it with a trapdoor on the top. Fly infront of the wild quetzal and make it fly inside. Unmount at a slight angle and the wild quetzal will get stuck. Now you can stand there mid flight and tame it.

Notice:

1 dont move your quetzal or the wild one could slip out, even when asleep.

2 add gates and a ramp to lure in ground animals.

3 gate to the front to trap smaller birds (just fly faster than them and "net" them like a jellyfish.

4 add ceilings around the trap as a platform to move on.

5 build the trapdoor and front gate in a way so they are open while you are flying (auto open and close).

6 add a ramp or path so you can get to your quetzal and mount it while flying.

Personally i added a fridge and bed to store kibble and let me spawn there.

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 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.~ Steel Cookie 123

If you’re on mobile and don’t mind spending a little bit of amber, kill one with a pteranodon or something [DON’T HARVEST THE BODY], grab its implant, create a trap around an obelisk or one of your revival platforms then revive it inside the trap and you can easily knock it out without using grappling hooks or making a flyer aggro it, risking your flyer being faster and killing it. A level 15 one would cost around 35 amber to revive though you can easily get amber by opening supply drops. This isn’t the best way as amber is hard to get but out of all the techniques I’ve seen it’s the easiest

(Old Ark Mobile)This is very long to do but effective, First tame a Tapejara, after doing that tame a high level female Rex make rex kibble. The the Quetzal taming, prepare, find a high level Quetzal, remember Tapejara saddles have two seats, switch seats then whistle 'target' then the Tapejara will follow the Quetzal, next shoot it with darts, when it's knock down apply soothing balm and put the kibble in it's inventory, make sure the area where it's knock down is safe.

The best KO method is to just chase it down riding a griffin, which is the only vanilla flyer that you can use weapons while riding. Alternatively you can "ride" a wyvern by standing on its back without actually mounting it and using whistle commands, you can also mount a tapejara's passenger seat and do that without the slight risk of falling like with a wyvern. Another alternative is to have a friend carrying you with a stamina leveled pteranodon (argentavis might work but it's a bit slow), but be careful not to accidentally shoot your pteranodon!

Simply get a snow owl, find a safe place, freeze it out of the sky,the quetz will walk for a few seconds.freeze again and place 4 dinosaur gates. Bring food and narcotics, enjoy your new quetz! This in my opinion is the Easiest way to tame a quetz in ark, no chain bolas no grappling hooks.