Yes, Ann/Agnes was from Ireland which is why it's been so difficult to find anything going back.
Is this the information you found for the 1851 census?
1851 Census: New Dundyvan, Old Monkland
Mary Mooney, head, W, 25, Derry, Ireland
John Mooney, son, 4, Old Monkland, Lanark
Mary Mooney, daur, 2, Old Monkland, Lanark
Patrick Hearty, Lodger, U, 30, Labourer, Armah, Ireland
Agnes Hearty, Lodger, U, 25, Servant, Armah, Ireland
Helen McDonald, Lodger, U, 20, Servant, Armah, Ireland
Bridget Connell, Lodger, U, 20, Servant, Galway, Ireland
Catherine McAea??, Lodger, U, 2, Old Monkland, Lanark
Agnes Gardner, Lodger, W, 30, Servant, Armah, Ireland
Thos. Gardner, Lodger, U, 9, Armah, Ireland
1861 Census: 27 Numbers Garth, Bishopwearmouth, Sunderland, England
Susanah Jary, head, unm. 21, house servant, Ireland
Mary Ann Jary, daur, 3mos. Durham, Bishopwearmouth
John Dogherty, lodger, marr, 20, tinner, Northumberland
Mary Dogherty, lodger's wife, marr, 21, Wales, Newport
Anne Tracy, lodger, Marr, 29, hawker, Ireland
Thomas Tracy, lodger, unm. 6, Durham, Sunderland (Thomas should be about 3 yrs)
Catherine Tracy, lodger, unm. 2, Durham, Sunderland
I've never found James Tracey (Ann's husband) on any census record.
Let me know If you find headstones for this family - I'd love a snapshot if there is a headstone.