The full sentence is 'Quintus no longer enjoyed his studies,' and I've translated it as 'Quintus non longior gaudebat studiorum.' Should 'studiorum' be genitive since it expresses possession?
2 Answers
To answer the main question, no—the owner goes into the genitive case, not the thing being owned. It's like the 's in English: you say Quintus's studies, with "Quintus" getting the special marker. So if you wanted to be very explicit about whose studies they are, you could say studia Quintī. You could also use the possessive adjective suus -a -um "their own": studia sua "his own studies".
But in this sentence, I don't think I'd use any sort of genitive or pronoun there. In English, almost every noun needs to have a determiner: a word like "the" or "a" or "his" or "that" that comes before it. Latin, on the other hand, doesn't. So while you can say specifically "these studies", or "his own studies", you don't have to; in many cases, I would just ignore the English determiner when translating, because the context already makes it obvious that these are Quintus's studies, not anyone else's.
You have a few other issues in that sentence, too. Non longior is "no longer" in the sense of "not having a greater length", not in the sense of "previously but not now"; I would recommend non jam instead. Gaudeō is closer to "rejoice" than "enjoy", so I'd replace that with something like fruor. And either way, the object of that verb should be in the ablative. But these are separate from the main thing you're asking about.
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2@MartinW Classical Latin had a few different instances where two sounds were written with the same letter; nowadays, people tend to use different letters for those, for clarity (i/j, u/v, a/ā, etc). For some reason i/j is less commonly written that way than the rest but I prefer to use it to make the pronunciation clear.– Draconis ♦Commented Jan 30, 2021 at 16:32
No. In your sentence, what are the studies supposed to be possessing?
The possessive case is used for those whom or that which possesses, and not are shown to be possessed. The possessive in your English sentence is his, not studies. You can think of the possessive case as the 's or "of" case in English: his, my, mine, yours, its, Quintus', of Quintus, etc.
Some other problems.
While you can used gaudere to indicate enjoyment, "to enjoy something" is better expressed as frui aliquo with the latter in the ablative. If this is homework help and you haven't learned that word yet, you should try to look at your examples properly using gaudere in this context.
Finally, "no longer" is best expressed by the phrase non iam (or similar). This is an idiom that must be learned.