Yes a garter snake bite does hurt at least in my opinion they do.
I once got bitten by a garter snake a few years ago and yes it did hurt and although garter snakes do release some venom into your body when they bite you the venom is not poisonous to humans.
So if you do get bit by a garter snake you don't need medical attention unless the bite is severe enough to need stitches.
When I got bit by a garter snake all I did was clean the bite up and use some hydrogen peroxide on the bite to clean it up and prevent further infection of the bite.
Then I just bandaged the bite and within a week the bite started feeling better.
The way a garter snake actually bites you is through repeated chewing so if the garter snake bites you once they'll bite you again if you do not get away from the garter snake in time.