I don't see it as a quirk. I think it's perfectly reasonable to not want to discuss business while on lunch break. But yeah, you need to be polite about it. I think the thing to do is let her finish her sentence (to see if it is, indeed, an emergency after all) and then, instead of addressing her comment/question, you let her know you'll address it later.
"I prefer not to discuss business on my lunch break unless it's an emergency. I can get back to you on that after lunch."
I think that you sort of already don't like this woman's work habits, as you noted, letting you know she sent you an email. Sometimes when someone you (general you) have a low tolerance level for does something that bugs you, you have a lot less patience than if it's someone you like more. I know I'm certainly guilty of that!
I think in your place I might go over and apologize.
"Sorry I snapped. I prefer not to discuss business on my lunch break unless it's an emergency and I sometimes forget that not everyone is aware of that."
And honestly, I don't see anything wrong with, in the future, when she comes over to tell you she's sent you an email, gently and politely reminding her that it's not necessary to make a point of telling you that, and that you will be getting to her email as soon as time permits.