Can it be like this? 不好意思。我可以借你的笔记吗
Zfe10's comment below seems to suggest that, yes, 借你的[noun] ("borrow your [noun]") is an acceptable (and perhaps even preferable?) alternative way to express 向你借[noun] ("borrow [noun] from you")
我可以向你借笔记吗 sounds weird, as a native speaker i wouldn't speak like that.
usually i'd just say 我可以借你的笔记吗？
What does the '向' mean in this context?
It is very difficult to understand this grammar
The "when", "where", and "how" information describing a verb must precede the verb in Chinese. "向你" is the "how" ("from you") and so must precede the verb "借" ("to borrow").
The grammar should otherwise be straightforward.