Kelly, I had never run across the unusual name Bilary, but I've tried to do a little census searching to see if I could establish a connection between our Carricks, without much luck I'm afraid. I found Steven (sic) G. Whalen and wife Bilary in the 1900 census. Bilary was said to have been born in Kentucky, her father in Virginia, and her mother in Kentucky. This was interesting, since my Carrick branch was in Virginia before migrating to Kentucky. I also found Steven Carrick in the 1870 census--this time his wife's name was listed as Billin (at least that's how it appears). Both ages match the birthdates you have. Let me know if you need more information from those censuses.
I then did a search of all the Carricks indexed in Kentucky for the 1850 and 1860 censuses, hoping to find Bilary as a five- or fifteen-year-old in one of those families before her marriage, but I had no success. The index may simply be incomplete, or perhaps her family moved out of Kentucky after she was born. Then either she alone or the whole family moved back later. Sorry I couldn't be of more help.