How can I recover Notes from an external hard drive

After you have gotten this sorted out, PLEASE begin using Time Machine immediately. There is NO REASON to not backup your computer daily!!!!!!!!!!!!!!!!

Back up your Mac with Time Machine

Navigate to that Mac's internal drive, and locate the following folder in the affected User Account:

~/Library/Group Containers/group.com.apple.notes/

Copy the entire contents of that folder to the identical location on the startup drive.

Migrating the contents of the startup drive that no longer boots was the right thing to do, but the reason your old Notes were not migrated is unknown.

You can find it in the same relative location. Navigate to the internal drive, then open its Users folder, which resides at its top level, specifically

/Users

You should be able to find the User Account within it. Open it, then open Library. The Group Containers folder should be within it. That's how you get to the location I described.

~ represents the fully qualified path to your User Account on that internal drive.

I understand neither User Account contains a Library folder. It's anyone's guess what may have caused them to disappear, but without a Library folder there is nothing to recover.

Their disappearance is almost certain to be related to the inability to boot the internal startup drive.

While within the user folder for your account on the internal drive, press the following three keys to toggle showing hidden items off & on:

Shitf  +   Command  +   Dot/Period

Normally the Library folders within user accounts are hidden from view and this key combination should reveal it if it still exists. You can confirm this is working by testing it by viewing your current home user folder within the Finder.

Which macOS version please?

Find out which macOS your Mac is using - Apple Support

The chance is rare after restoring the Mac. Don't pay for the data recovery software as it is almost useless in such a case!