I know we call our moon “la lune,” but what if one is generically referring to “a moon,” such as “a moon of Jupiter”? Is it just “une lune”?
Yes, "lune" is also used in French both as the name of Earth's satellite and as a synonym of "satellite".
I would add that, when referring to our moon, the correct writing is:
with an uppercase L. When referring to other moons as regular satellites,
would be the correct way to write it.