How can I restore data from Notes to my new laptop?

Here is how to use that backup to migrate to a new Mac:

… Transfer to a new Mac with Migration Assistant - Apple Support

This works best when the migration is the starting point for the new Mac.

if you’re already using the new Mac, this gets… more involved.

Moore L wrote:

I am already using the Mac. The more complicated instructions would be much appreciated.

Option one: Save off what you have changed, wipe all of the new stuff and re-install macOS, and migrate the whole environment. Restore your changes.

Option two: create a new login on macOS, migrate your content in there, and manually transfer the notes between the new and your existing login, and then delete the new login.

Option three: create a new login on macOS, migrate your content there, and manually transfer your changes from your existing login to into your newly-restored environment, and delete your old login.

If you follow option three…. You will likely have also picked the exact same login name from your old environment for your new, and that is the login name that you want to use in your new environment. Which means a login rename. These instructions work, if followed exactly:

Change the name of your macOS user account and home folder - Apple Support

Here is the intro to Migration Assistant, which is how content gets moved around

Transfer to a new Mac with Migration Assistant - Apple Support

Have backups of all of your data prior to starting any migration.

The easiest option was option zero, initial migration when starting with the new Mac. That ship sailed.

Of the other options… When you migrate, you can select what you want to migrate. I’d usually migrate everything relevant, because I’ve never encountered a Mac with nothing added other than local notes. Which option really depends on how much work will be involved to merge your data—and this merge is going to be manual—and that depends on how much content and context you’ve added to and have customized on the new Mac that doesn’t exist on the old Mac.

If you did actually have only notes, and if remote storage was used for those notes, then there’s an option four: enable iCloud Notes or the mail server where Notes are stored by the old Mac on the new Mac.