This has always been a frustrating feature of the application, but yes notes do not transfer when the lead is qualified.
You can write code to do this but I might use it as an opportunity to become introduced to a great ISV solution from North52 Business Process Activities. They have a free version of their solution that will support this customization without writing code. Warning: it may also introduce you to other powerful capabilities that might entice you to subscribe!
The process you're trying to create is laid out in an example they even publish in their KB. It's XRM Formula #142.