Sit on the Tapejara,set that Quetzal as target and go to second or third part of saddle,now you can easily knock out Quetzal with tranq darts or tranq arrows.
Don't knock out Quetzal over Red Woods or swamps,I lost here 4 Quetzals.They are spawning in Mobile nearby red obelisk,that is best taming location.
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.