Antimos’s name day is September 3, the feast of Anthimos of Nicomedia.
Churches that follow the Old (Julian) Calendar keep the same feast on September 16 of the civil calendar. See everything commemorated on September 3.
The saint behind the name

Anthimos of Nicomedia
September 3, Hierarch, Ανθιμός
St. Anthimos was Bishop of Nicomedia during the great persecution under Diocletian at the opening of the fourth century. When many of his flock were slain, he shepherded the survivors from hiding, strengthening them by letters and prayer, until he was discovered, confessed Christ before his judges, and was beheaded.
Also celebrating on September 3
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.