Gizmo - 2015-01-01

You can do it by setting the G Keys as keyboard keys.... such as v for PTT and Num5 as mute

However I second the option to have G-Keys setup so they can be used natively with mumble

It has been done with other software

The first link is an example of a native implementation in teamspeak
and the second link is the developers section of the logitech site where the GKey SDK can be found

If I have time I'll have a look myself as I just bought a G930

(I am just a mumble user, with enough programming experience to be dangerous so don't expect results :))