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.
Another time money comes up is the episode, In the Cards, where Jake wants to buy a Willie Mays baseball card to cheer up his dad. He talks Nog into paying for it because as a Federation citizen Jake doesn't have any money.
I found some of the dialogue on Memory Alpha:
It's my money, Jake! If you want to bid at the auction, use your own money.
I'm Human, I don't have any money.
It's not my fault that your species decided to abandon currency-based economics in favor of some philosophy of self-enhancement.
Hey, watch it. There's nothing wrong with our philosophy. We work to better ourselves and the rest of Humanity.
What does that mean exactly?
It means… it means we don't need money!
Well, if you don't need money, then you certainly don't need mine!
I also imagine Starfleet officers in a non-Federation posting getting stipends. Or maybe Quark and other business owners bill the Federation for whatever Federation citizens buy? And maybe that arrangement doesn't extend to the auction with the baseball card?
There are references to Jake buying from some of the station's businesses. I think I remember him buying jom-jom sticks, and dining at the Klingon restaurant. (Did Jake ever use a holosuite on his own?) I'm thinking either Jake has some sort of allowance or stipend, and is exaggerating when he says he "doesn't have money", or those businesses make special arrangements for Federation citizens.
This leads me to another thought: are station residents charged for using the replicators in their quarters? I imagine the replimat charging like other station businesses; so what about private replicators? Or maybe I'm wrong, and the replimat is a free public service?
I'm guessing replicator use would be free under Federation jurisdiction the same way Quark isn't charged rent. But on the other hand, it's a Bajoran station - Starfleet only administers it. Now I'm wondering what kind of negotiating went on to get the struggling provisional government to pass up revenue from DS9 rental spaces.
You just know Bajor happily relinquished the potential income from rental spaces on the station in exchange for the Federation keeping their military presence there.
I’m thinking either Jake has some sort of allowance or stipend, and is exaggerating when he says he “doesn’t have money”
I mean, having walking around money and having Mom-style "now that's walking around money" (i.e. enough to buy a Willy Mays rookie card) are two different things.