Actually, the answer is simple. The QMC is an interface to the QlikView Management Service. That service is running under a different account than your Browser is.
Since Mapped drivers are user-specific (i.e. they aren't assigned globally for all users on a Windows machine) your mapped drives aren't visible to the QMS account.
Because of this, and as an administrator, you should try to use the actual UNC paths wherever possible. For example, if you create a document on your local machine that accesses a shared file, use something like:
FROM '\\servername\share\MyExcelDocuments.xlsx' (...);
instead of Q:\ (mapped drive). In that way, your document will work out-of-the-box on your server, provided that the service account has access to the same shares on your network. No need to mess with your server configuration anymore.
BTW, in QMC the best thing you can do is copy/paste the complete UNC path in the target field. There is a Folder Dialog, but the Network Browser part at the bottom of the tree in that dialog doesn't always show all machines on your network.