Can and could are both fine here, though I would recommend adding ever to clarify the intended meaning (unless other context already makes that clear):
- "Can happiness ever bring pain?"
- "Could happiness ever bring pain?"
The difference between the two is that can seems to be asking about whether it ever really happens, whereas could seems to be asking about whether it could even hypothetically happen. (Neither of these is strict, however; we can say things like, on the one hand, "Is there any possible world where happiness can ever bring pain?", and, on the other hand, "Do you think that, in the average person's life, happiness could ever bring pain?")
[…] the sentence is meant to ask whether there are possible conditions under which happiness bring pain. Or, does it ever happen that happiness brings pain.
But those aren't the same! For the former, I'd recommend could; for the latter, I'd recommend can (or does).