Agathe’s name day is February 5, the feast of Agatha.
Churches that follow the Old (Julian) Calendar keep the same feast on February 18 of the civil calendar. See everything commemorated on February 5.
The saint behind the name

Agatha
February 5, Martyr, Αγάθη
St. Agatha was a virgin martyr of Sicily who suffered for Christ during the persecution of Decius in the third century. Of noble birth, she consecrated herself to the Lord, and when she refused the advances of the pagan governor she was subjected to cruel tortures, which she endured with unshaken faith, dying in prison from her sufferings.
Also celebrating on February 5
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.