In general I would suggest introducing someone in around seasons 3-7. Stay away from the Klingon political ones. If she's not 100% into sci-fi there are some excellent episodes to choose from to break her into it slowly.
Here's some of my favourite episodes.
Yesterday's Enterprise, Qpid, The Pegasus, Attached, Parallels, All Good things...(Though I wouldn't suggest starting at the end), Darmok, Conundrum, The Inner Light and Timescape.
I've deliberately left out season openers/finales.
If you're girlfriend is already into sci fi then you have a lot more options. If not then I'm afraid it's going to be trickier but a good character driven story from the series should do the trick. Personally I would go with The Inner Light of the ones I've mentioned.
Hope that helps. I am well aware that this is not an easy task!