Judith Finlay's husband and children's names sound Catholic, so I don't think you'll find them in Protestant church records like the Finlays I've been tracing. Do you know if that's the case, or what other denomination they were?
Catholic marriages didn't usually record the bride and groom's fathers' names and didn't become matters of civil registration requiring this until 1864, but Protestant marriages from 1845-on were government registrations and have not only the couple's ages, status (single, widowed, etc.), and residence, but fathers' names and occupation. Very helpful for genealogy.
LDS Search Pilot of Irish Civil Registrations has only one marriage for a Judith Finlay in Co. Monaghan:
Name: Judith Finlay Registration district: Castleblayney Record type: MARRIAGES Registration date - quarter and year: 1871 Film number: 101251 Volume: 11 Page: 474 Digital GS number: 4179384 Image number: 00401
However, there are 1860s marriages for other Finlay women who might have been your ancestor with a double name (Judith Ellen, Judith Alice, Judith Catherine, etc.). You can search these on the LDS web site: