In TNG, Picard says that the Federation has evolved past a need for money. Indeed, we never see any.
In DS9 though, Quark talks a lot about bar tabs and costs. Surely O’Brien and Bashir don’t get free drinks, so how do they pay? I’d assume that any Ferengi worth his lobes won’t accept anything that can be replicated, so do Federation officers get a stipend of tradeable “value” when interacting with cultures that still expect payment?
I think there’s also a reference to Quark paying rent to Sisko for running the bar. Presumably that’s denominated in latinum. I wonder where it goes? Maybe the secret “Garak black ops” fund.
They probably do, since the Federation deals with civilisations that use money on the regular. They likely have an allowance/stipend specifically for that.
“Farpoint Station” has Dr Crusher ask for some cloth to be put on her tab on the Enterprise, so DS9 isn’t the only station using a similar system, which might be the same system, it’s just that Quark’s is a Ferengi business on a Bajoran station, so the issue comes up more often compared to someone who might be living on a starship, or on Earth Spacedock.