I would assume Quark speaks Federation Standard

I assume he made a clear point of never learning it.

I'd put larger odds on Nog and Rom eventually learning it (but not at the time of the screenshot).

One thing I've always wondered, and this is hardly the most unbelievable aspect of the science fiction elements of this show but it's one of the most pervasive and constant; why don't we ever hear the native languages of the aliens underneath the sounds of the universal translators? Most, although not all, of them speak by vibrating the air, so where are those sounds disappearing to?

Quark would never lower himself to speaking anything human. He views humans and the Federation as inferior and foolish.

I read a theory once that "daimon", the Ferengi title for captain, could translate to "good/lead merchant" or something along those lines.

So if we assume that's true, it's possible that "hewmon" means something like "shitty merchant", and it's just pure coincidence it basically sounds like human.

Human is the one word they know in Federation Standard. They go out of their way to mispronounce it.

That's what I figured

Makes sense considering we know from Darmok that the universal translator doesn't translate proper names.

It's not even head cannon, this screenshot is legit not just from the episode, but from the scene where there's an example of exactly what you described.

For one word? Everything else is fine. That's not really how accents work.