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.
If your on crystal isle and your soloing it here's a good trick I used, first tame yourself a moth get a couple of parachutes and grappling hooks and your preferred amo with weapon, once you found one you like fly next to the quetzel and hop off the moth and grabbling hook it (dont forget parachute incase) and whistle it to attack quetzel the moth doesnt do damage but instead will slow it down for it bit while you shoot at it from below. This helped me tame it without having to whistle passive, attack target repeatedly.