If Query Store is enabled on a given database, when that database gets backed up, the information within Query Store will also be backed up. No extra steps are necessary to make this happen. It's just a part of the database, like any other table, procedure or view. The page-by-page copy that a BACKUP database command does, includes the Query Store data.
Now, if you're not seeing Query Store data, it's likely one of three things. First, you're not actually using a BACKUP/RESTORE process to move the databases around, but instead are using some type of import/export or something else. Second, Query Store is not enabled on that database, so there is no Query Store data in it. Third, someone is running the command to clear the Query Store data either before the backup is taken, or after the restore is processed.
Those are the mostly likely causes from the info provided.