Passports per se aren't needed to NI (or even the Republic), but usually some form of photo-identity is for air travel.
Best to phone or email the airline to see what is needed - I bet they'll ask for a driving licence!!!!
Luton to Belfast sounds like easyJet. If so, "children under 16 years of age travelling on UK, Italian, Spanish, or French domestic flights can travel without photo identification provided that they are travelling with an accompanying adult": http://www.easyjet.com/EN/Book/regulations.htm l#infants
i.e. since you're travelling on a UK domestic flight, your baby does not require any form of photo ID.