Maksim’s name day is January 21, the feast of Maximos the Confessor.
Churches that follow the Old (Julian) Calendar keep the same feast on February 3 of the civil calendar. See everything commemorated on January 21.
The saint behind the name

Maximos the Confessor
January 21, Hierarch, Μάξιμος
St. Maximos the Confessor was a monk and theologian of the seventh century. Leaving high office in Constantinople for the monastic life, he became the great champion of the truth that Christ, being true God and true man, possesses both a divine and a human will, standing almost alone against the Monothelite heresy supported by emperor and patriarch.
Also celebrating on January 21
These names share the feast, many of them other forms of the same name:
Maximos, Makis, Max, Maxim, Maximus
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.