Did the ancient Athens have military culture?
Yes, to an extent they did but their society was not as heavily militarized as say, that of Sparta. The center of a Greek citizen’s life was his polis (the city state he belonged to) and defending it in times of need was considered a sacred duty. The Athenians maintained a part time army made up of citizen soldiers, usually co-commanded by politicians who rotated command. The citizens were well equipped and reasonably trained but they were not true professionals. The Athenians put much more stock in their navy and maintained a large fleet crewed by professional rowers. You should google the battles of Marathon and Salamis to learn more about Athenian military prowess.