The subscription site Roots Ireland does not have any baptismal records in all of Ireland for a Connolly baptism with father Thomas and mother with the surname Heffern. Likely the church where he was baptized no longer has records back to 1839.
Have you located the 1880 US census for Patrick Connolly in Rochester and are there any clues with that record? How long did he live? the 1900 census may also be helpful.
Have you considered DNA testing?