How can I diplomatically inform her that I am rejecting their project due to her partner's attitude?
If you must mention the reason behind your refusal, then consider the following:
Thanks for taking the time to discuss project XYZ with me and introducing me to your business partner. While the project is interesting, during the meeting with your partner I found that our style of communication doesn't mesh well, and I don't believe I'm a good fit for this role in your team. Thank you for the opportunity, and good luck in your endeavor!
It conveys what you want - that the issue isn't with the project, but with the existing team - without speaking negatively of any individual. Obviously she doesn't have this issue, so calling him out as a jerk, or someone with a bad attitude isn't her experience, or if it is she's learned how to transcend such communication patterns and still get work done.
Even if it is a significant problem, or if he's truly a jerk or has a bad attitude, you don't necessarily want to call it out, partly due to your limited experience with him, but also because it's not your place to ruin their business relationship.
Do I even need to mention about his attitude, or just say no to her without any further justification?
You do not need to mention the real issue. If she digs for a reason then you may want to bring it up, depending on your level of familiarity with her, but there's also the issue where she may be bringing a project to you for your benefit. She may not actually need you, and it is possible that her partner is unhappy with the situation and was unable to completely hide it during the discussion.
Thanks for meeting with me about your project. I'm not going to be able to accept it, but I wish you luck with it!
There's no need to volunteer more information. If she really, really wants you to work on this project she will press for more explanation, and after a discussion it's possible you may come to an agreement you can both live with (that involves less contact with the other person, perhaps).
If she's doing you a favor she will drop it immediately, unless she's a close friend and is really interested in helping you out financially - then she'll probably press for more info, but directed towards your financial state and what you plans are to rectify it.