Joseph’s marriage to Mary Ann Nicholson and Joseph was registered with the civil registrar in Lurgan April – June 1884 Volume 1, page 857. If you obtain the civil cert you should get both father’s names, occupations and addresses (ie more information than is on the religious certificate). Joseph’s name is spelled CABREY on the Familysearch site.
There’s a birth for a Charles Carbery in Armagh City 1866 Volume 1, page 63, also William John Carbery in Lurgan 1871 Volume 11, page 713. Can’t say whether these are the right births but they broadly fit the information you have given.