For example, there's someone at work that I really don't like, and she's good at her job. She makes me want to work harder and outdo her. Can I say, "She really motivates me"? To me, it sounds a little like she motivates me in a good way when it isn't. Is there another word that I can use? Or can "motivate" be used here?
I'm not sure if there's a word that combines annoyance with motivation, although I'm sure someone will post it if there is! But you could just say that wanting to beat her motivates you, or something along those lines - it's the competition that inspires you, or the outcome you're looking for.
There are words for this kind of relationship though, like rivalry. Also the word drives has a less positive feel than motivates - it's not necessarily negative, but it can be associated with something that controls you and pushes you in a certain direction.
I feel that having "She" as the subject of the sentence isn't going to work. "She" isn't doing anything, except being there and doing her job. Instead it is "I" who is finding motivation in this situation.
It sound like you are competing with her. You want to beat her, you could also use "better" as a verb.
I'm really competitive, and I just want to beat her.
If you prefer to use "She" as the subject:
She makes me competitive...