Ann’s name day is December 9, the feast of the Conception of the Theotokos by Anne.
Churches that follow the Old (Julian) Calendar keep the same feast on December 22 of the civil calendar. See everything commemorated on December 9.
The saint behind the name

Conception of the Theotokos by Anne
December 9, Great Feast, Σύλληψη Θεοτόκου υπό Άννης
On December 9 the Church commemorates the conception of the Most Holy Theotokos by the righteous Anna. Childless for many years, Joachim and Anna prayed with tears, and according to tradition an angel announced to each of them that God would grant them a child who would be spoken of in all the world. Anna conceived and bore Mary, the future Mother of God.
Also celebrating on December 9
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.