For Adam Carson B 1833 +/- 4 yrs, somewhere in Ireland, with the father's forename unknown, I found 1 rcd, 1834, Co. Antrim. Father's forename was John.
For James Carson B 1833 +/- 5 yrs, ditto, I found 13 rcds - too many to easily summarize here. However, 5 were in Co. Antrim. None had a father forenamed John.
The forenames and surnames don't sound Gaelic-Irish. Thus, they were likely protestant. B/M/D rcds are sparse in norther Ireland, especially early protestant ones. The results found above could be due to pure random chance.