Currently we are hosting a customer's instance in our Amazon EC2 Account. We would like to move this to the customer's account so that the billing is transferred.
Is there an easy way to simply migrate an instance to a different Amazon Cloud account?
No, you cannot move an running instance from one account to another.
You can however create an AMI of that instance and share the AMI with the other account. Take note of "Sharing an AMI with Specific Users":