@kjrocker do you have a sample project which can replicate this for me?

Basic connection info:
https://cloud.google.com/sql/docs/postgres/external-connection-methods
And specific to Google App Engine:
https://cloud.google.com/sql/docs/postgres/connect-app-engine

That page has a callout which says:

App Engine Standard only supports connections to PostgreSQL from the following runtimes:
Python 3.7, Java 8, PHP 7.2, Go 1.11, Node.js

But @dazuma has a tutorial which connects in this same way:
https://cloud.google.com/community/tutorials/elixir-phoenix-on-google-app-engine

Access the production database from App Engine
The App Engine runtime also runs a Cloud SQL Proxy for you, and makes your databases available via Unix sockets. In the App Engine environment, these are located in the directory /cloudsql at the root of the file system. So, to prepare your app for deployment into App Engine, edit your config/prod.secret.exs file again, and modify the socket_dir database setting to point to the correct location for App Engine...
Note that if you need to run another Ecto migration or open another psql session from your local workstation, you can temporarily revert socket_dir to /tmp/cloudsql so that Phoenix can talk to your local Cloud SQL Proxy. If you do, make sure you change it back to /cloudsql before you deploy to App Engine. Alternatively, if you have the ability to create the directory /cloudsql on your local workstation, you can configure Cloud SQL Proxy to open its sockets there instead, and avoid the need to revert socket_dir.