Some people put their garlic in the fridge for about 2 weeks. You aim to get a temperature of about 4 - 6 deg Celcius, so your fridge might not be cold enough. Some people have reported that garlic stored in the freezer sprouts just fine. It may be that they used hard neck varieties which are more suited to the cold since they originated in the cold climate of central Asia. Soft neck varieties were developed from the hard neck. However, garlic is normally planted below the frost line outside.
When garlic is in the ground before winter, it does grow roots which help prevent frost heave but that is not going to be an issue for 2 weeks in the fridge.
Budding doesn't need cold temperatures though as garlic planted in spring does bud too in temperate climates induced by the increasingly longer days, but you'll end up with smaller bulbs by the time it's finished.
When people grow garlic inside, they do that to eat the greens and not grow bulbs. To grow bulbs inside you're going to need either a sunny window with all day sun or artificial lighting. And you'll need a pot that is about 1 foot deep.
Garlic can also be grown all year round in temperate climates. We normally plant the cloves on the shortest day of the year and harvest on the longest day. I planted mine late last year in late spring into pots and pulled them out in late summer. The bulbs were just a bit small, though. And I had a problem with onion aphids which may because I used a potting mix high in nitrogen.