Solo taming is easiest with a tapajara, who often get stuck on cliffs and are easy to tame. Get into the jara front seat, whistle attacking target on the Quetz with a spyglass, then whistle around with the "move here" command. Remember to whistle passive before getting to close the the Quetz. Just don't hit your jara with tranq darts or it will fall asleep and you will fall out of the saddle from very high off the ground.
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.
In mobile, one method is to shoot it with a dodo arrow(stone arrow and two dodo feathers)and lure it into a trap, but that is not recommended because it doesn’t stay aggroed for long. This is for those who can afford it. Dodo feathers only come from dodos with a size of 1.55, and the chance for them to drop is very low, making enough ammo hard to get.
If you manage to knock a quetzal out, try starve taming. If you never tried starve taming it’s just leaving a Dino’s hunger go to half way or lower and then you start feeding. I’ve done this many times and it’s very useful taming trick.