For someone who doesn't already have a Quetz, the best way to tame it is to have a buddy pick you up on either a high stamina Ptera or fast Argent and wait to line up the tranq shot against it's head. Ideally the more of these the better but don't be waiting too long for a good shot otherwise the torpor will quickly go down. When you think you're close to finishing make sure the last few are shot when it's not above water so it doesn't drown, then finish with either a circle of spike walls or some kind of box for protection.
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