Debbie,
For someone born c 1821, there are no birth certificates. They only started in Ireland in 1864. You have to reply on church baptism records. Not all churches have records for the 1820s and of those that do, not all are on-line.
You don’t say what denomination Anna Maria was but looking at the 1831 census of Co. Londonderry there were about 10 Hood households and 9 were Presbyterian and 1 Church of Ireland. (Presbyterian usually indicates Scottish origins normally in the 1600s. Half the population of the county are of similar descent.). The 1831 census only names the heads of households so neither Anna Maria nor Ellen would be listed themselves.
http://www.census.nationalarchives.ie/search/
Does Londonderry mean the city or the county? If it means the county, then that’s a lot of church records to look at. If it means the city then there are about 3 or 4 churches with records for the 1820s. There are copies in PRONI (the public record office) in Belfast. If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
You mention that Anna Maria may have gone to work for a Protestant Minister near Fair Head. That’s the parish of Culfeightrin. There’s a Church of Ireland in Ballyvoy and another in Cushendun, but somehow I can’t see them still holding records of a servant who worked there 180 years ago. I think this will be a hard family to research without more information.