No one should smoke weed.
Should my girlfriend let me smoke weed?
No She should not do that.
It isn't a case of your girlfriend letting you do something. It is your choice at the end of the day. You need to consider if it's the right thing to do. Personally I would say that you shouldn't smoke weed, it's a massive waste of money and can do you no good. But, you need to make that choice for yourself.