QBMP files are just backups of the original files, much like the QB2007. So should you need to "convert" the data, you could simply rename the files. But always be careful with handling these files, as they can easily overwrite your current data.
Convert QuickBooks backup file to QuickBooks 2007 for Mac backup file.
We have found one software suitable in our database for this conversion.
Apple macOS / Mac OS X software - convert qbmb to qb2007 on OS X
QuickBooks for Mac is financial manager used to organize finances in one place, track money going in and out of business etc.
Convert from qbmb file
to qb2007 file
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.