Nick’s name day is December 6, the feast of Nicholas the Wonderworker.
Churches that follow the Old (Julian) Calendar keep the same feast on December 19 of the civil calendar. See everything commemorated on December 6.
The saint behind the name

Nicholas the Wonderworker
December 6, Hierarch, Νικόλαος ο Θαυματουργός
St. Nicholas the Wonderworker, Archbishop of Myra in Lycia, lived in the fourth century and is one of the most beloved saints of the Church. Renowned for secret almsgiving — according to tradition he saved three daughters of a poor man from ruin by gifts of gold given at night — he was a defender of the innocent and of the true faith in the age of the First Ecumenical Council.
Name days: Nikolaos, Nikos, Nikolas, Nikola, Nicolas, Nicholas, Nick, Nikoulas, Niko, Klaus, Claus, Cola, Kola, Nikoleta, Nikoletta, Nicoletta, Nicole, Nikolina
Also celebrating on December 6
These names share the feast, many of them other forms of the same name:
Nikolaos, Nikos, Nikolas, Nikola, Nicolas, Nicholas, Nikoulas, Niko, Klaus, Claus, Cola, Kola, Nikoleta, Nikoletta, Nicoletta, Nicole, Nikolina
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.