Antony’s name day is January 17, the feast of Anthony the Great.
Churches that follow the Old (Julian) Calendar keep the same feast on January 30 of the civil calendar. See everything commemorated on January 17.
The saint behind the name

Anthony the Great
January 17, Venerable, Αντώνιος ο Μέγας
St. Anthony the Great, the father of monasticism, was born in Egypt in the middle of the third century. Hearing in church the Gospel call to sell all, give to the poor, and follow Christ, he gave away his inheritance and withdrew into the desert, where he spent the rest of his long life in prayer, labor, and fierce spiritual struggle.
Name days: Antonios, Antonis, Tony, Antonio, Anton, Antony, Anthony, Antonia, Tonia
Also celebrating on January 17
These names share the feast, many of them other forms of the same name:
Antonios, Antonis, Tony, Antonio, Anton, Anthony, Antonia, Tonia
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.