I don't think it's incompatibility, but I wonder if she is nervous if you are putting on a front in the relationship. I wonder if she is trying to reconcile that you seem one way when you work, and another when you two are together? It's possible that she could be concerned that you would adopt this same attitude in your relationship, especially since you have been dating a good bit and you two are really investing in each other. Or maybe that you already have this cocky attitude and are concealing it, only to release it into the relationship when you feel comfortable. If you know your industry and are just conveying what you know then you aren't being cocky, you are being confident. Now you need relay this to her.

Maybe you could explain it like this the next time it comes up:

"I understand that you feel like I am cocky professionally. I feel that I have a deep understanding of my industry, and part of building up my brand and thus my income is to let people know that I have this understanding. I think it's fine to be honest with people about my abilities and to sell myself in the process. I hope you know that I don't view our relationship like this, I value collaboration. I want to work together to figure out what is best for us."

Or something that fits for you. Hopefully by now you have a track record of acting collaboratively, and it could be good to gently remind her of times you have worked with her to figure out the best course for the two of you. Maybe explaining it as your job and your relationship both access different parts of your personality is a better way of explaining it, saying that there are two different people might not help you right now. I get what you mean though.