When I was growing up I remember the most common pronunciation around me in North Hertfordshire was GARR-ij. But my impression is GARR-ahj is gaining ground and I think I use both pronunciations nowadays.
The OED offers a third British English variant, guh-RAHJ which seems fairly close to the French word garage we borrowed, which is from garer 'to cover, protect'. (This verb survives into modern French but today means 'to park').
The OED stops there, but other sources stick their etymological necks out a bit more. Wiktionary traces a mongrel ancestry through Old Frankish and Old Norse doublets *warjan and *varask respectively, both thought to be from a Proto-Germanic root *warjana- to do with covering, protecting and so on. This in turn is traced to a Proto-Indo-European root *wer- with the same meaning.
If this is true, which seems quite probable, garage has a series of interesting cognates, all sharing some sense of being over something or protecting it. For example
weir, warn, guarantee, garrison, warren, garment, garnish, garret
It's even possible pert is cognate. This is more surprising, but the derivation is thought to be from a prefixed form *ap-wer-yo‑ which would mean uncover or open. This ended up in Middle English as apert, which got shortened to pert and changed its meaning, via the sense of open, to 'saucy' or 'forward'.