RC church registers in Ireland aren't centralised or computerised either in County Cork or County Dublin/Dublin City and their availability and how they might be accessed depends on the town/townland of birth. Cork had nearly 6,000 towns/townlands.
There are no civil registration records in Ireland before 1864 (Protestant marriages in 1845) so there is no way in Ireland to find when or where either she or John Brown were born. You need to know the specific birth location in Ireland, i.e. town/townland, civil parish and county of birth from their and their children's US records.
If she was born in Dublin, you would need to know whether it was County Dublin or the City of Dublin. If County Dublin, you need to know which of the nearly 1200 towns and townlands. If the City of Dublin, you need a street addy to find which of the hundreds of churches might have her baptismal record and where and how the registers might be accessed. You also need to know her parents names.
There wouldn't be any information about her or her marriage in the US in Ireland. That would be in US records so if you believe she was married in either Louisiana or New Orleans you would need to search for a marriage record there -- either a civil or church one. You could also post a request for information about her possible marriage there on the Louisiana and New Orleans Message Boards.
There's no information in Ireland about who emigrated, where they went or what happened to them after emigration. That information is in the locations where they lived after emigration. So any relationship between John Brown and Mary would need to be determined through their civil and church US records.
Her first born and subsequent children's civil and church birth, baptism, marriage and death records may include where Mary was born in Ireland.
You need to continue research on Mary and all her children in all their US records which I listed in order to obtain the information you need to begin researching in Ireland.
Sponsors at various events such as marriages and baptisms may have been relations from the home area of Ireland and you can research their census, civil and church records in the US also.