After you’ve tamed your first quezt, on PvE, ask another tribe (you provide the resources) to craft a good saddle if you aren’t high enough level yet. Pump nothing but stam into your current one, build a quetzal trap and let the desired tame fly in. Close the trap, knock it out in the air, and you’re good to go. Make sure you bring lots of stimberries and meat so the quetzal doesn’t run out of food or stamina.
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.