Do I need a permit for garage conversion in Unincorporated Monroe County?

Most likely yes. Garage conversion usually requires a permit in Kentucky. Confirm the specifics with Unincorporated Monroe County.

Kentucky does not require a building permit or inspection for a single-family dwelling unless the local government has passed an ordinance creating a residential inspection program (KRS 198B.060 and 815 KAR 7:125, Section 2(2)(a)), so the first question is whether the city or county covering the address has one. New wiring in the converted space needs a state-required electrical permit, and added fixtures need a plumbing permit, whether or not a local building program exists (KRS 227.480, 815 KAR 20:050). Every sleeping room needs an operable emergency escape and rescue opening with a sill no more than 44 inches above the floor (KRC Section R310.1), and any remaining garage may not open directly into a room used for sleeping (Section R302.5.1). Kentucky's residential energy provisions follow the 2009 International Energy Conservation Code (KRC Section N1101.1), so ask the local office what insulation it expects for the converted space. Turning a garage into a bedroom, living room, office or apartment changes it to habitable space, which must meet the code for living areas, such as insulation, heating, ceiling height, light, and an emergency escape opening in any sleeping room. A conversion normally needs a building permit, and often electrical, mechanical and plumbing permits too. Zoning may also require you to keep off-street parking or limit a second unit, so check zoning before you design.