Well, I'll assume you already destroyed or finished converting all buildings out of Hellenistic culture? You also don't have any "likes Greeks" generals in the province proving +2/4 Hellenistic culture right under your nose?
There is only one other province and I doubt they have 2 temples of Zeus so you should be gaining Germanic culture soon, you may need one more temple in Sparta though to offset their culture influence per turn else you may be stuck on stagnant (no loss no gain, just an offset). But no, you don't need a whole province to gain culture, you just have to be pumping out more culture on your buildings and characters than the other factions in the province.