Serge’s name day is October 7, the feast of Sergios and Vakchos.
Churches that follow the Old (Julian) Calendar keep the same feast on October 20 of the civil calendar. See everything commemorated on October 7.
The saint behind the name

Sergios and Vakchos
October 7, Martyr, Σέργιος και Βάκχος
Saints Sergius and Bacchus were high-ranking officers in the Roman army at the end of the third century, secretly Christians and beloved of the emperor. When their faith was discovered they refused to sacrifice to the idols; stripped of their rank, they were paraded in mockery and cruelly tortured. Bacchus died under the blows, and Sergius was afterward beheaded in Syria.
Also celebrating on October 7
These names share the feast, many of them other forms of the same name:
Sergios, Sergey, Sergio, Sergius
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.