I have a Treanor/Trainor/Traynor database for Kilbroney parish which covers Rostrevor and this is what I have on your family:
8 Feb 1875 James Trainer married Ann Gribben in Kilbroney parish. Witnesses : Hugh Morris, Rose Berry
Children of James and Ann :
Born 4 Jan 1876 Mary, daughter of James Trainer, Ann Gribben.
Born 24 Jan 1877 John, son of James Treanor, Ann Gribben
There may have been other children but I could not find them.
In 1901, James and Ann are living in Knockbarragh townland, outside the village of Rostrevor with their daughter Mary. In 1911, James, a widower, is living in Knockbarragh with his daughter Mary but in 1914, Mary marries Joseph Murphy in Rostrevor Catholic Church on 19 May 1914.
Here are possible deaths for James and Anne :
name: Ann Trainor
registration district: Kilkeel
event type: DEATHS
registration quarter and year: Jan - Mar 1906
estimated birth year: 1849
age (at death): 57
volume number: 1
page number: 487
name: James Trainor
registration district: Kilkeel
event type: DEATHS
registration quarter and year: Oct - Dec 1920
estimated birth year: 1847
age (at death): 73
volume number: 1
page number: 368
You can find them in 1901 on this website :
http://www.census.nationalarchives.ie/pages/1901/Down/Rosstrevor/Knockbarragh/1236998/Their surname is spelt Treanor in 1911.
Unfortunately I do not have the father of James Treanor so I cannot go further back in my database. If you order the marriage cert for James Trainor and Anne Gribben, that will give you their father's names. Then repost to this site and I will see If I can get you back a further generation.