Agreed with what KevC already posted, and just to further clarify: The database in question simply is not part of the file system that you can typically see through your HTTP or FTP access to your site, and therefore isn't part of most normal types of backups folks do of "their site files".
If you ever used the "Maintenance" tab in ACP and invoked "backup", selected "all items" and "store file locally", then the /store/ folder of the phpBB site might have one or more of these database backups that you could restore. But that's still not "the live database", and will only be as current / up to date as the date of whenever that backup had actually been invoked. But even if that backup is months or years old, it might be your only database backup.
If what you actually have is some kind of hosting-specific backup image of your site (e.g. on GoDaddy.com currently, there is an "Installatron" site installation and backup system), then its possible there could be a restoreable database backup contained within that backup package. Since you described just copying the file system manually to your hard drive, this is unlikely to be an option in your case.
For taking the backup copy of your site to a new host, and getting the database and config.php correctly setup for the new host so that you can restore any database backup you might be able to come up with, the knowledgebase article "Knowledge Base - Transferring Your Board to a New Host or Domain
" discusses the kinds of steps that need to be performed or considered when doing this.