According to this
document, Cloud SQL instances reside in their own parent VPC network, and it is not possible to move the instance to a different shared VPC service network. However, you can configure the Private IP with the shared VPC service network on the Cloud SQL instance at creation time. That said, the private services access must already be configured for the network and setting the Private IP will only work when you are creating a new Cloud SQL instance with the
gcloud beta command.