Na they don't have a separate economic wing for this. They understand what drives everything in the system. Its labor. If the Federation needs something from the Ferangi, they have labor to trade. They send technicians to perform some form of high end labor. This would be like, high end system design, automation design, energy optimization, lawyers and legal design, terraforming and habitat restoration, or large scale construction.

There are probably a lot of redlines however, which makes them a hard negotiator. The wage consortiums like Farangi know that they can fall back on the federations unlimited asset supply, and can negotiate for how ever many self sealing stem bolts as they need. They also know not to crash their own economy with this asset, and very likely the federation knows they hold the power to totally crash any wage economy. It's probably baked into the Prime Directive.

I mean they canonically converted that capitalist they unfroze into the ambassador to Franginar! He didn't run off to become some leader of a wage consortium. He became an ambassador to Franginar because he had real life capitalist experienced instead of historical theoretical understanding of capitalism.

So I think they just provide people labor for trade. They also likely keep stocks of latinem and they do it by periodically trading labor for latinem or more likely commodities they replicate for latinem.

Most currencies are going to be digital in Star Trek. Gold Pressed Latinem is just the asset that backs up the currency used by Farangi. Quark likely keeps his own physical latinem on hand, and uses it on the dabo wheels for the novelty of it, since 99% of his patrons are federation or bejorin. The crew must have some way to shore up the tab at quarks and I think they do that with labor. Sure, some of them probably hold Latinem for when they don't want to fix stuff at quarks, but they must also perform labor and provide logistics for quark to make the relationship square for him. He likely doesn't pay rent (I can't recall) or utilities, and does have to put up with some form of federation regulations. The trade off is that the federation members get to eat and drink and socialize for free.

I'm sure I could mine all sorts of episodes to build this sketch out, but that must be how their trade works. I once saw a video that suggested, based on evidence from the show, that each crew member only worked at most 4 hours a day, maybe less to maintain the functions within the enterprise. They are so incredibly productive that they'll likely have lots of time to dedicate to other projects and tasks. This leaves a lot of time to perform some exchangeable labor to support the federation's inter system trade.

I always liked to imagine the federation had a separate economic system for trading with capitalist coded races in Star Trek.

This is head cannon? I thought thats just how things worked since The Federation never seemed to have a use for it other then trading externally.

You pretty much nailed it, it's not even headcanon, it was confirmed in DS9 https://www.youtube.com/watch?v=lSW8WndZcYU