Birth certificates only started in Ireland in 1864 so you won’t get one for Martha. You might find her baptism. RC records are on-line on the nli site. For other denominations eg Church of Ireland, Methodist, Presbyterian etc you will probably have to go to PRONI in Belfast.
She married in Seagoe Church of Ireland which suggests she was that denomination. The PRONI catalogue lists these records for Seagoe:
C.I. Seagoe (Dromore diocese)
Baptisms, 1672-1731, 1735-1821 and 1829-76;
marriages, 1672-1731, 1735-1821 and 1826-45;
burials, 1672-1731, 1735-1821 and 1829-81; vestry
minutes, 1734-1900.
Analysis of register of baptisms, marriages and burials,
1672-1904.
Index to baptisms, marriages and burials, 1672-1919.
Preachers’ books, 1877-; visiting book, 1879.
3 Hall households in Kernan in 1901:
http://www.census.nationalarchives.ie/pages/1901/Armagh/Kernan/Kernan/Probate abstract for a possible relative:
The Will of James Hall late of Kernan County Armagh Farmer who died 12 March 1893 was proved at Armagh by Andrew Hall of Lisniskey in said County Farmer the surviving Executor.
The will itself is on-line and names his wife as Ellen.
There is a John Hall in the Valuation revision books for Kernan (on the PRONI site). He had plot 20, a farm just under 12 acres. He is deleted in 1870 (suggesting he died around then). He is replaced by Thomas Hall & James Hall jointly on 20a and Malcolmson Hall on 20b.
There is a death for a John Hall registered in Luragn 30.3.1869 aged 75. That could be him. You can view the original certificate on-line on the GRONI website, using the “search registrations” option:
https://geni.nidirect.gov.ukYou will need to open an account and buy some credits. It costs £2.50 (sterling) to a view a certificate.
Thomas looks to have been the son of Moses Hall. His bride, Mary Jane, was another Hall also from Kernan, her father was Isaac. Thomas & Mary Jane married in 1869:
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1869/11408/8189446.pdfHere’s Thomas son of John, from Kernan marrying in 1854. He was a pensioner indicating he had probably been in the army. The John Hall is a weaver not a farmer, so keep an open mind in case it’s a different John.
https://civilrecords.irishgenealogy.ie/churchrecords/images/marriage_returns/marriages_1854/09474/5433518.pdf