Translation:Can I borrow the notebook from you?
The meaning is a little bit different. A notebook means you have some notebooks and I may just need one. In Chinese, the quantity should be specify, if u wanna "a" notebook, you say 我可以跟你借"本"筆記本嗎. Secondly, if say so, it sometimes implies give me something and I won't return it.
In this sentence, the notebook it means your notebook. I borrow and I'll return it.
"The notebook" sounds weird and wrong in English unless it happens to be a specific notebook that was already referred to earlier in the conversation, which may still be one of several owned by the other person; it implies "the notebook that has already been mentioned".
If the Chinese can't be interpreted as "a notebook" (which I'm not convinced of), then "your notebook" makes more sense, absent any other context.
In the context of a language lesson, I'm guessing formal grammar rules win out: http://www.quickanddirtytips.com/education/grammar/can-versus-may
的 after 你 would be used in cases of ownership, but there is no ownership implied in the sentence here. In the exercise above, 你 is in the sentence as part of 向你 (towards you/ from you). As in, "Can I borrow the notebook FROM you?"
我可以 = Can/may I 向你 = from you 借笔记本 = borrow the notebook