This sounds rather odd in English. Usually, I would talk about the culture of Greece. Were I to talk about the culture in Greece, it sounds to me like I might be talking about Albanian culture or Syrian culture or something located in Greece.
In modern greek, it sounds weird to put the noun before the adjective. But ελληνικός πολιτισμός sounds better, yeah. This phrase emphasizes the place. Ελληνικός πολιτισμός could be referred to another placesthat ancient greek culture exists. As example, in modern Turkey.