BIRMINGHAM family Cork

I am looking to see if I can get more information on my grandparents who were from Cork.

My GGrandmother Mary Elizabeth was born somewhere around 1883-1888 in Cork and married Alfred William SMITH in 1906 in Hampshire. On her marriage certificate it shows her father as a John Joseph occupation Army Pensioner.  I have found in the 1911 Irish census a John Joseph b 1839 occupation retired army pensioner living at 10.3 Lady's Well Hill in Cork.

This unfortunately is the only information I have.  I am hoping to be able to trace this line back further.  If anyone is researching this line please contact me.

Thank you 

Nicci