The main problem with doing something like this is that Gmail creates a new message every time your note is updated.
Here's an example:
Create a new note, with the text "Test". This is what Gmail looks like:
1: [Notes] Test
Update this note to say "Test Updated". Gmail now looks like this:
1: [Notes] Test Updated
2: Test
Update it again to say "Test Updated 2". Gmail now looks like this:
1: [Notes] Test Updated 2
2: Test Updated
3: Test
This means that if you do get forwarding to work (which there is at least one way I can think of to make it work), it will be a nightmare to deal with all the duplicates. The only way to prevent duplicates, is to never update existing Notes. You must either delete them immediately after you create them, or get in the habit of treating them as read-only. If you choose the former, you may be better off using the Evernote app or Mail app instead, both of which involve less steps.