How to wire this dimmer switch
The green wire does need to connect to one of the two grounding screws in the box. 

To completely comply with code practice, you would only place one wire under each grounding screw, but unfortunately the previous electrician has cut off the ground wires of the existing cables too short to pigtail a wire connector to them. You could try loosening the cable clamp and try to pull the cables far enough into the box to provide a minimal ground conductor length (if careful, you may only need 2-3 inches), wire nut them together, and connect one conductor to one of the grounding screws. (Don't forget to retighten the cable clamp.) Then connect the green wire from the dimmer to the other screw. This arrangement allows the dimmer to be removed without opening the grounding connection to the rest of the circuit. 

If neither of the cables has enough slack outside the box to do that, then you should still run the green dimmer wire under one of the two grounding screws, but it's not a best practice.