Welcome to Ireland Reaching Out!
The Griffith's Valuation for 1856 mentions a George Fletcher here:
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNam…
From this starting point you could have a look at the PRONI Cancelled Books to see if the Fletchers are named. The Books for Pomeroy run to 1929. You can find more about them here:
http://www.irish-genealogy-toolkit.com/revision-books.html
The PRONI images are easy enough to access but you will need a bit of time to go through each digitised book. Other parishes are also accessible.
Rootsireland.ie has a baptism for a Bridget Fletcher born to a father George and a mother Bridget McKenna in Killey/Killea, Pomeroy and baptised on 31 Jan 1838. Other siblings records include Ann was also baptised on 2nd April 1840, Rose baptised on 20th Nov 1842, Betty (possibly your Elizabeth) baptised on 3rd Sept 1845 and Sara Jane baptised on 14th August 1850. This would make George quite aged by the the standards of the time by the birth of Sara Jane, approximately 60-65 years old - though not an impossibility!
You could also check up rootsireland.ie (a subscription site) or familysearch.org for further children and grandchildren.
Hope this gives you some pointers.
Best wishes
Clare Doyle
Genealogy Support