Mokios’s name day is May 11, the feast of Mocius the Martyr.
Churches that follow the Old (Julian) Calendar keep the same feast on May 24 of the civil calendar. See everything commemorated on May 11.
The saint behind the name

Mocius the Martyr
May 11, Martyr, Μώκιος
The Holy Hieromartyr Mocius was a priest who suffered for Christ at the beginning of the fourth century, during the persecution of Diocletian. Preaching boldly against the worship of idols, he was arrested and endured many torments, and was finally beheaded at Byzantium, faithful to the end.
Also celebrating on May 11
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.