You can export the QB2006 to QBB using the File → Backup to QuickBooks Windows function in QuickBooks for Mac. Be careful when restoring older backups, as they might overwrite your existing data. Also, it is not possible to use new version of the format in the previous version. Ie. you cannot open *.qbb file exported from *.qb2012 in Quickbooks 2011 etc.
Convert QuickBooks 2006 for Mac backup file to QuickBooks backup.
We have found one software suitable in our database for this conversion.
Apple macOS / Mac OS X software - convert qb2006 to qbb on OS X
An OS X version of popular accounting software
Convert from qb2006 file format
The table with program actions contains information about what each program is capable of doing with their files. This may be a good pointer for further searches for certain file conversions and help you find the converter you are looking for if our search result is insufficient for you. However, it is far from perfect and may sometimes show results which are not really usable because of the nature of the software handles the files. So a conversion that appears as possible, just because of matched actions, may in fact not be possible.