There seems to be some syncing issues. I can open a file in the drive app, but I can't open the same file directly in the Quick Office app. I can see it, but tapping it does nothing. Probably because it was just a note file. However, if I create a Word file in Quick Office, I can long press it in Drive and remove it and is disappears in Quick Office.
What kind of file are you trying to delete? Quick Office can only view Word like documents, spreadsheets and presentations. It will see all Drive files, but it can't open them.