Anthony Farquhar was born and raised in South Belfast and educated at St. Malachy's College. He later studied Classics at Queen’s University, Belfast. He was then sent to study Theology at the Pontifical Lateran University, Rome. He was ordained to the priesthood on 13 March 1965, aged 24.
He was Assistant Priest at Dunsford and Ardglass Parish. He was a hospital chaplain in 1966. He was later assigned to teach Latin at St MacNissi's College, Garron Tower. where he stayed as a teacher from 1966 to 1970. From 1970 to 1975 he was Assistant Chaplain, along with Rt Rev Msgr Ambrose Macaul ay, to Queen's University, Belfast and finally as Chaplain/Lecturer at the University of Ulster from 1975 to 1983.
His main outside interests are folk music and football, particularly student football. He is a very active president of Queen's University Belfast A.F.C. and also serves as patron of the Irish Universities Football Union.
On 6 April 1983, aged 42, he was appointed Titular Bishop of Hermiana and, with Canon Patrick Walsh, as Auxiliary Bishop of Down & Connor. The Principal Consecrator was the Bishop of Down and Connor, Cahal Daly, (now Cardinal Archbishop Emeritus of the Archdiocese of Armagh) and Principal Co- Consecrators were Archbishop Gaetano Alibrandi, the Apostolic Nuncio to Ireland and Bishop William J. Philbin (Bishop Emeritus of Down and Connor). His Episcopal Motto is 'Sapientia Proficere' ('to increase in wisdom') from Luke 2:52. He is currently Chairman of the Irish Bishops Conference on Ecumenism.
He was one of the Bishops that travelled to Rome for their Quinquennial Visit Ad Limina visit in October 2006. In December 2007 he officiated at the dedication of the new Altar in the Church of the Good Shepherd in his native Parish of the Most Holy Rosary, Belfast, following the renovation programme.
A new book titled Inter-Church Relations: Developments and Perspectives has been published by Veritas to mark the 25th anniversary of his ordination as a bishop. In it, a number of distinguished contributors have praised Farquhar's work, including the Sean Cardinal Brady who notes that he was “engaged actively in promoting bonds of friendship and understanding at times when it has been far from easy to do so”.
On December 3, 2015 Pope Francis accepted his resignation on the grounds of age. He was at that time the longest serving bishop in Ireland.
- "Bishop Anthony Farquhar, the longest serving bishop in the country, resigns on age grounds". Diocese of Down and Connor. 2015-12-03. Retrieved 2016-02-01.