Artemios’s name day is October 20, the feast of Artemon of Seleucia.
Churches that follow the Old (Julian) Calendar keep the same feast on November 2 of the civil calendar. See everything commemorated on October 20.
The saint behind the name

Artemon of Seleucia
October 20, Martyr, Αρτέμιος
The holy Great Martyr Artemios was a distinguished military commander of the fourth century who served under St. Constantine the Great and his successors. When the Emperor Julian the Apostate sought to restore paganism, Artemios rebuked him openly for his persecution of the Christians; for this he was stripped of his rank, cruelly tortured, and beheaded at Antioch in the year 362.
Also celebrating on October 20
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.