If you already have a quetz you can build a Contraption on the back of the quetz platform saddle. If you do not have a quest already and easy it is to use the tapejara, other than that use a ptera or a argy and follow it everywhere and tranq it down make sure you have spikes and narcotics ready, fast torpor!
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.
Easy way : Bring a harpoon gun with net ammo and BASICALLY Shot the quetzal and make sure it falls in a safe area and then trap it as fast as you can and Tranq it with your ammo of choice and Normal Gates can work and even if it wakes up it shouldn’t be able to escape easily and to net it again you have to De-agro it