Also as much as I hate hate to say it, if her Trek knowledge is zero, J.J.'s Star Trek 2009 movie hooks a lot of people. Then you ease them into the real stuff
Believe it or not, Star Trek '09 is real stuff too.
My advice is not to try to get her hooked actively, that means don't get her to agree to watch a certain number of episodes, don't overthink the issue by picking the best introduction episodes etc., just watch episodes you like when she's around. She'll notice that you're watching a certain show and that you're enjoying it, eventually she will plop down on the sofa and watch it with you, you'll be surprised how much she already picked up just by walking through the room.