Can I change the network of an existing Google Compute Engine VM

You have to recreate the VM's in the new network.

Google Developers Console

Although you cannot directly move an instance, the Google Developers Console now has the function to Clone this instance. Go to Google Developers Console, navigate to Project ID -> Compute Engine -> VM Instances -> . Scroll down to the end and you should see the option

After clicking on Clone this instance, you will be brought to the new instance creation page, with an option to select which Zone you wish to clone the target instance.

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gcutil

You can use the gcutil moveinstances command to move an instance from a zone to another zone. You can even move a list of instances using regular expression, for example:

gcutil moveinstances --source_zone=zone-a
                             --destination_zone=zone-b "i-[0-9]" "b-.*"

For more information, type gcutil help moveinstances

Just create a snapshot of the vm instance, then create a new vm instance from that snapshot and specify a different network.