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Birth records are kept at the local level so you must know location AND they are protected by privacy legislation for 110 years (unless you are a direct descendant).

Interestingly, there is another family with the same name in the 1891 Census in Wales; perhaps a connection worth pursuing.

The name of her father should be noted on the marriage record from 1952 in Wales.

James played for Glenavon FC in Lurgan in the years prior to his marriage:

https://www.national-football-teams.com/player/38781/James_Burns.html


https://nifootball.blogspot.com/2006/08/james-burns.html

I know you have this info, but posting for others in case it provides some small clue for other researchers.

If George was born in Dromore (perhaps 1928) you won't find a birth record for him online altho' you could check the GRO.  I note that altho' James was playing for a Dublin team, Mary seemed to have been a resident of Dublin, unless she met him elsewhere and followed  him to Dublin.  Where did James play prior to Dublin?  Have you done your DNA to look for Martin matches?

The records for the Berkeley Street Church (St. Joseph's) are held in local custody.

https://registers.nli.ie/parishes/0512

Using the search function available on A***try for the 1911 census, these are the options for a Mary Martin born around 1904 with a father named William:

Monkstown Farm, Kingstown #3, Dublin -- Catholic

Windsor Avenue, Templecorran, Co. Antrim (noted as Mary Ritchie Josephine)

Ballintemple, Blackrock, Co. Cork -- Catholic

Woodvale Road, Shankill, Belfast (noted as Mary Kathleen) -- Presbyterian

Cableroad Street, Templecorran, Co. Antrim (noted as Maria Dennison Martin).