Patrick Flannigan and Elizabeth Coate born around 1817 Ireland

Patrick and Elizabeth (Eliza) are my Great Great Grandparents who came to England about 1846 and settled in County Durham around the Trimdon, Coxhoe, Kelloe area. I can trace them in Durham well but there is no indication of where they came from in Ireland and I would love to find out. Eliza's maiden name is unclear and it is given as "Coate" "Cute" and "Coote" in the records. Similarly the spelling of Flannigan varies in the records but settles out as Flannigan eventually. Their first daughter is Mary Jane who isborn in Ireland in 1845. Their second daughter is Ann (Annie) who is born in County Durham in 1847 followed by Ellen (b1853), and my Great Grandfather Owen (b1856) also known as John. I can see from where they have their Baptisms and funerals that Patrick and Eliza were Catholic. Patrick dies in 1890 in East Hetton and Eliza in 1895 and they are both buried at St Helen's Church, Kelloe, County Durham. If anybody can give me some clues of where they may have come from or how I can find out, I would be very grateful

Martin