I wonder how this would work if one parent lived in the Federation and one lived, with the child, in a capitalist society.
Would they be out of luck? Would the Federation pay out on the absent parent's behalf(we already know they have reserves of other currencies, and also have "Federation credits" for trade with capitalist societies)?
The UFP has a currency - Federation Credits. The average citizen may not need (or even have) money, but it exists and is presumably accessible in some way if you have genuine need. For example the station personnel on DS9 never had issues paying at Quark's, so one can assume that they did get a salary because they were posted to a capitalist environment. I'd assume that child support would work on a similar "as needed" basis.
Feddits appear to be a stable currency at first until the chief administrator disappears in the neutral zone and is rumoured to be a prisoner on Romulus. Then the federation needs to restart feddits as feddit.org.