Methodius’s name day is May 11, the feast of Cyril and Methodios, Enlighteners of the Slavs.
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

Cyril and Methodios, Enlighteners of the Slavs
May 11, Apostle, Κύριλλος και Μεθόδιος
Cyril (baptized Constantine) and his elder brother Methodios were born in Thessalonica in the ninth century. Sent by the Emperor and Patriarch Photios to Moravia, they devised an alphabet for the Slavonic language and translated the Scriptures and services, so that the Slavs might worship in their own tongue.
Name days: Kyrillos, Kirillos, Cyril, Kyril, Methodios, Methodius
Also celebrating on May 11
These names share the feast, many of them other forms of the same name:
Kyrillos, Kirillos, Cyril, Kyril, Methodios
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.