Do I need a permit for garage conversion in Globe?

Most likely yes. Garage conversion usually requires a permit in Arizona. Confirm the specifics with Globe.

Converting a garage to living space changes its use and usually adds wiring and cooling, so cities, towns, and counties that enforce a building code generally require a permit. Arizona has no statewide energy or residential code, so the insulation and egress rules come from your local code. If the garage becomes an ADU, ARS 9-461.18 and ARS 11-810.01 make ADUs a permitted use on single-family lots but leave building codes in place, and bar requiring extra parking for the ADU. Arizona has no statewide residential building code. Cities and towns adopt their own codes, and a county board of supervisors may adopt a building code for zoned unincorporated areas and may exempt areas zoned rural or unclassified (ARS 11-861), so the permit rule depends on where the property sits. 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.