The green (ground) connection to the grey (presumably neutral) is wrong, just like any neutral-ground bond anywhere other than the main panel. If the grey wire were to slip out of the wirenut, the body of the transformer would rise to 120 volts (hot, through the transformer primary to the white wire to the green wire) and could electrocute someone!
Look for a wire, or a bundle of wires, either bare, green or green/yellow. This is the ground. Connect the transformer green wire to that (If a bundle, add the new wire to the wirenut, don't separate the existing wires).
If there are not such wire(s) in the box and the box is metal, you should connect the green wire to the box. There should be a screw hole tapped for a 10-32 screw. Use a 10-32 x 3/8" machine screw to secure the wire to the box. You may already see a screw securing the existing ground wire: do not remove it. Look for another suitable hole/screw.
If the box is plastic and there is no sign of a ground wire in the box, just put a small wirenut over the end of the green wire and tuck it into the box.
Edit based on the photo and additional information:
I see several problems now. Whoever installed the old transformer cheated quite a bit. First, the transformer black wire connected along with another wire to the breaker ( it's generally not allowed to put two wires under a screw). Second, even though white and green wires are being connected to the ground/neutral bus, you still aren't allowed to use a single wire to connect these.
Disconnect both wires from the breaker and add a short length of #14 black wire as a pigtail. Connect one end to the breaker and connect the other end to the other two wires with a wirenut.
Disconnect the transformer green wire from the grey wire and use another pigtail (green) between it and the ground/neutral bus. also, if the grey wire is smaller than 14 gause, you should replace it with a #14 white pigtail.