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.
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:
If the answer is the right solution, please click "Accept Answer" and kindly upvote it. If you have extra questions about this answer, please click "Comment".
Note: Please follow the steps in our documentation to enable e-mail notifications if you want to receive the related email notification for this thread.