Best solo method that I've found, by far, is to use a Archaeopteryx with a Tapejara. Once you find a Quetz, jump off and switch to front passenger. Whistle attack while shooting tranqs, then whistle passive once you get too close. I bring Argies and set target and follow distance both to high, make them follow the Tapejara. When you eventually knock the Quetz, whistle the Argies to follow you to defend it wherever it lands. Make sure all the Argies are set to "Only attack conscious".

Any taming food or kibble I bring, I place in the Tapejara and have them set to passive when I land. Once you knock a Quetz out of the sky, you can use crouch to go straight down with the Tapejara so you don't lose where the Quetz lands.

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