Those green connectors are TomTom connectors. They don't fit much else, and they provide 12V to the Tomtom with no step down. If you are going to connect it to a non-Tomtom GPS, you'll need to step it down from 12v to 5V.
I made an adaptor cable by buying the matching Tomtom connector (it's also used in some cars, like a few Mercedes), and then slicing that Tomtom plug into a Garmin lead. Effectively I made a connector cable that has the Tomtom plug at one end, and the miniUSB connector at the other, with the voltage stepdown in the middle. Only thing to watch for is the polarity.
Of course, you can always just remove the existing Tomtom connector and wire the Garmin bits to it directly, but I wanted a piece I could move from bike to bike, since several Guzzi models use the same Tomtom connector.
Eric