If you have avocados that are not fully ripe yet then no you should not put them in the refrigerator to ripen.
Putting unripe avocados in the refrigerator will slow down the ripening process of the avocados.
It's best to leave the avocados out at room temperature to ripen as the avocados will ripen faster at room temperature.
Once the avocados ripen at room temperature then you can put them in the refrigerator.
There's really no other way to ripen avocados other than giving it some time and letting the avocados sit out at room temperature.
The avocados should ripen at room temperature within 4 to 7 days but if not they should ripen a few days later.