You don’t say what denomination the couple were but I assume it was RC. RC marriages weren’t recorded in the statutory records until 1864, so you won’t find a statutory marriage for this couple. If you knew where they came from you could search the local RC parish registers, but obviously you don’t have that information. You could try Rootsireland but not all parish records are on their site, so it’s no comprehensive coverage. (And some parish records are lost too).
I assume the couple, or their children, left Ireland. Usual advice is to search all sources looking for clues as to where they came from eg censuses, marriage & death certs, wills, obituaries, gravestones, military records and Poor Law applications (if they lived in England or Scotland). Otherwise it’s a needle in a haystack.