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.
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
Fly on a peteradon above a Qetsil and jump of with a perashoot but us two gusn wit darts to traqalise it but make sure your bird is on folow en make sure it is on high so as soon as the Qetsil flys away jou can get back on your peteradon and you can follow the Qetsil
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.