Naoum’s name day is December 1, the feast of Nahum the Prophet.
Churches that follow the Old (Julian) Calendar keep the same feast on December 14 of the civil calendar. See everything commemorated on December 1.
The saint behind the name

Nahum the Prophet
December 1, Prophet, Ναούμ
The holy prophet Nahum is one of the twelve minor prophets of the Old Testament. He prophesied the fall of Nineveh, the proud capital of Assyria, declaring that the Lord is slow to anger and great in power, a stronghold in the day of trouble to those who trust in Him, and that the oppressor's city would not rise again.
Also celebrating on December 1
These names share the feast, many of them other forms of the same name:
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.