With James and Mary being such common names in Ireland, anything is possible, but I don't think there's a connection between our families. My O'Flaherty's haled from Wexford in the late 1700's and early 1800's and my 2x Gt GF emigrated to NZ in the early 1870's. I've also checked my tree and I do have a Kate born 1857, and a John born 1859, quite a lot earlier than the dates you mentioned, plus I don't have any O'Connor's in my tree.
Sorry I couldn't help this time. Good luck with the rest of your search.