Hi @JC ,
Welcome to Microsoft Q&A!
Your problem does not seem to be related to sql server, but rather more seems to be related to Google Cloud.
I tried to solve your problem and here are the links that I think might help you.
https://stackoverflow.com/questions/56503352/how-do-i-connect-to-my-google-cloud-sql-instance-with-c/56503446
https://stackoverflow.com/questions/52457709/how-to-connect-google-cloud-sql-with-c-sharp
Also, is it connection name or is it public IP address I use for server name?
Server = "Google Cloud SQL IP address"
Here is the Google Cloud Support i find from google:
https://support.google.com/googlecloud/?hl=en
Best regards,
Seeya
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