Steve is, as usual, correct.
I'm on vacation and out of the country, but it looks like there are battling forms of cleverness here and they're defeating each other.
MainWindow::dropEvent() _should_ be putting the exact name into inputFileNameText via the setText(). The explicit quoting in MainWindow::browseInputFile() is quite suspicious. We're calling QProcess, not a shell, so we shouldn't be in the business of performing shell quoting to handle whitespace in pathnames. If we're going to support multiple names, we should probably make inputFileNameText take a QStringList and quit trying to stuff everything into a single quoted string. Once we do this, MW::applyActionX() probably needs some attention when getName() returns a string array.
On the other issue, if the output filename isn't an absolute path, we should probably call http://qt-project.org/doc/qt-4.8/qdesktopservices.html#storageLocation
to find the Desktop (DOcuments?) directory and then canonicalize that dir + path.