John,
I have found what looks to be the birth for Mathew Fitzgerald. The birth was registered in Merthyr Tydfil, Glamorgan in Jan – March 1857, Volume 11a, page 285. Merthyr is about 5 miles from Rhymney and is likely the civil registration town for that general area.
I did not find a birth for Hannah in Merthyr Tydfil but there is one registered in Abergavenny that is possibly her. Abergavenny and Merthyr are fairly close and so possibly that’s where the family were living in 1854. That part of South Wales was a big coal mining area and miners moved around a fair bit. (I assume the Fitzgeralds will have gone there for work. Thousands of Irish migrants did the same).
The birth that might be your Hannah was registered in Abergavenny, Monmouthshire, Jul – Sept 1854, Volume 11a, page 90.
You can order copies of those birth certificates from GRO in Southport, Lancashire. I think they cost £9.25 (sterling) each (USD $11.30 or thereabouts). See:
http://www.gro.gov.uk/gro/content/
They usually take about 2 to 3 weeks to arrive.
If the Abergavenny birth is not your Hannah, then the next nearest match was Ann Fitzgerald born Merthyr Tydfil July - Sept 1853, Volume 11a, page 2591. So that’s exactly 1 year earlier than the information you have. Ann and Hannah can be interchangeable, so I would give it consideration. (The page number seems very large and may not be correct but that’s what’s on Ancestry’s index).