Logically the only thing I can think of is you are using the old WordPress database and only deleted old files from the old site. BPS Pro files MUST exist somewhere or it would not be possible for the Quarantine email alerts to be sent.
Possible scenarios/options:
Reinstall and uninstall BPS Pro on the new site. Uninstalling BPS Pro removes all files and all database options. Deleting BPS Pro files manually would not remove database options for BPS Pro.
Use either phpMyAdmin or the Adminer WordPress plugin and edit your database. Do a search for bulletproof_security and delete all BPS Pro database options, but like I said, in order for AutoRestore/Quarantine to be able to send email alerts the BPS Pro files MUST EXIST somewhere.
Unless of course these emails are backlogged and they are just now being sent. ie you install a new site and emails that were queued are now being sent.
Another possibility is that you have BPS Pro installed under another website on DesktopServer and you used the “Add Additional Files/Folder” option in AutoRestore to monitor another website. BPS Pro can monitor ALL files anywhere under an entire Hosting Account and also ALL files under a local Development environment.
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