You don't have to give any reason at all for quitting. You have a contract of employment, and if you follow the notice procedures you're within your rights.
Personally, I would always aim to stay on the best terms with former employers. I can't imagine not wanting my co-workers to know what I'll be doing next. And as you're not heading to (or starting up) a competitor, there's every reason to want to make this move totally in the open.
Your concern seems to be not the leaving, but the fact the two of you are leaving together. So what are the risks?
The most obvious one would be if you have been working at your startup on company time. Have you? If so, now is going to be the time to face the music. And if not, being transparent now saves you from whispers down the line.
Assuming you have a good relationship with your line manager, I would be inclined to talk about this face-to-face and in private. She's unlikely to make problems for you if you show how excited about and committed to your new venture you are, and if you also show how committed you are to leaving well - tying up loose ends, handing over your role and generally recognizing that until you leave, she has the first call on your time and energy.
Leave well. You never know who you may find yourself dealing with again in the future!