Leontos’s name day is June 18, the feast of Leontios.
Churches that follow the Old (Julian) Calendar keep the same feast on July 1 of the civil calendar. See everything commemorated on June 18.
The saint behind the name

Leontios
June 18, Martyr, Λεόντιος
The Holy Martyr Leontios was a soldier of Greek descent who suffered for Christ at Tripoli in Phoenicia in the early centuries of the Church. A man of courage and charity, he confessed the Lord before his accusers, and tradition tells that his steadfastness under torment brought even those sent against him to faith in Christ, so that they shared his martyrdom.
Also celebrating on June 18
These names share the feast, many of them other forms of the same name:
Leontios, Leonidas, Leon, Leontius
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.