Konon’s name day is March 5, the feast of Konon.
Churches that follow the Old (Julian) Calendar keep the same feast on March 18 of the civil calendar. See everything commemorated on March 5.
The saint behind the name

Konon
March 5, Martyr, Κόνων
St. Konon was an early martyr from Isauria in Asia Minor, remembered as a man of simple and holy life who drew many of his countrymen from idolatry to faith in Christ. When persecution reached his homeland, he confessed the Lord openly and endured torture at the hands of the pagans.
Also celebrating on March 5
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.