Each of your 5 cables should have a bare copper wire along with the black and white wire. All 5 of those bare coper wires along with the two green wires from the fan and light should be joined together with one big wire nut. If your box was steel (it is not) you would add another short piece of bare copper wire into the large wire nut and attach the other end to the box with a green screw.
Connecting the two green wires together wont accomplish nothing. If the cables coming in dont have the bare copper wire then there isnt much you can do. In the old days that wire wasnt required. By code you need that wire or you need the power supplied to the circuit to be GFCI protected. Many homes are still wired without a ground wire and everything works fine they are not as safe as they can be and not to code. This room is your kitchen and there are a few other branch circuits running from this box and you are not sure where they go. If they go to outlets close to the sink and such that require GFCI you should think about replacing those outlets as well if they are the regular outlets now.