I've converted far more people over to Star Trek using DS9. Unlike TNG you can show them the pilot without risk of prematurely scaring them away. That said, if you're determined to use TNG, there are a lot of good episodes to choose from. Really depends on what kind of person your girlfriend is, if she enjoys action sequences, character drama or comedy material. But a few episodes you can't go wrong with in my opinion..
Measure of a Man
Most of these are strong episodes that have little or no backstory that needs explaining beyond who everyone is.