Use a Tapejara, land near the Quetzal and get on the front seat. Use LB + down arrow and make the Tapejara fly to your command. You'll fly faster on attack target just make sure to click another area before it bites the Quetzal. This method does not use any of the Tapejara's stamina so you only need to worry about weight.
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 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.