Well to be clear, the Green River ends at the Confluence, inside Canyonlands National Park, which is quite a ways from Moab.
More to the point of the original post, if you want to just keep moving and not hike or explore, then Green River, Utah to the Confluence (where the Green meets the Colorado) is a great flatwater float. You can do it in the time you've outlined, and then pay $100 for a jet boat to take you back up the Colorado to Moab.
There is a required permit if you are doing Labrynth/Stillwater (the trip mentioned above). It's available at the Green River State Park office (one put-in option) for free.
There are a number of shuttle options for vehicles and boats. Google those topics in connection with Green River and Moab, and you'll have your pick. Here is the one I'm most familiar with, in Moab.