I'm not sure what you mean by "transfer the contents of the old computer to this new one", but yes, if you indeed transfer every file from the old computer to the new, the files should all be there. They might be 'hidden' in some way, or just buried deep in the hard drive (it won't be enough just to transfer the Documents, for instance), like:
C:\Users\Martin Degrell\AppData\Roaming\Thunderbird\Profiles
Are you not able to download the MozBackup tool to a USB drive or CDR, install it on the old computer, make the backup, but the backup on the USB/CDR, and transfer it to the new computer? Seems a bit easier than to lug the entire contents of the old machine onto the new - though I'm guessing you have other stuff that needs transferin'.