In 1952, Fr. Markey was ordained a priest for the Diocese of Clogher in Ireland.
In 1981, Fr. Markey came to Minnesota to participate in a Clinical Pastoral Education program at Willmar State Hospital.
In 1981, Fr. Markey worked at St. Joseph in Henderson. In 1982, Fr. Markey worked at St. Andrew in Granite Falls for about three months, after the pastor there died and before a new pastor arrived. Fr. Markey left the diocese in June 1982.
In 2012, Fr. Markey died in Ireland at the age of 84 while awaiting trial for sexual abuse of a minor in Ireland in 1968.