You can upload your Google Docs files right to Moodle for assignments. When you click the "Add..." button for submitting a file, just choose the Google Drive repository. You'll need to log in. Please use your college account for Google Drive.
Once you choose your document, it will get copied into Moodle. Changes you make to your document after you submit it will not be reflected in the copy! Very important. You can't submit a Google Doc and then keep working on it.
Handing in a Google Doc as a Word or on a Turnitin Assignment? See here for Exporting from Google Drive