Matthew’s name day is November 16, the feast of Matthew the Evangelist.
Churches that follow the Old (Julian) Calendar keep the same feast on November 29 of the civil calendar. See everything commemorated on November 16.
The saint behind the name

Matthew the Evangelist
November 16, Apostle, Ματθαίος ο Ευαγγελιστής
The holy apostle and evangelist Matthew was a tax collector at Capernaum when the Lord called him with the words Follow me; he rose at once, left everything, and made a feast for Christ in his house. One of the Twelve, he wrote the first of the four Gospels, setting forth Christ as the fulfillment of the Law and the Prophets.
Also celebrating on November 16
These names share the feast, many of them other forms of the same name:
Matthaios, Mattheos, Matt, Mathew
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.