Glykeria’s name day is May 13, the feast of Glykeria.
Churches that follow the Old (Julian) Calendar keep the same feast on May 26 of the civil calendar. See everything commemorated on May 13.
The saint behind the name

Glykeria
May 13, Holy Woman, Γλυκερία
The Holy Martyr Glykeria was a young Christian woman of Thrace who suffered for Christ in the second century. The daughter of a Roman official, she confessed the Lord openly when the pagans of Heraclea gathered for sacrifice, refusing to honor the idols. For this she endured imprisonment and many tortures, and was finally given over to wild beasts, ending her contest with prayer.
Also celebrating on May 13
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.