EDIT 17th August 2019: Added instructions for the newly added consent screen below..
Recently I came across the need to connect to Google Sheets with a secure authentication process quite often, so I will share with you how and to what extend I got the custom connector working that I found here. It uses OAuth2 authentication, so you can share your workbook with selected colleagues and they will be prompted to enter their credentials in Power BI if they try to access these files.
High level overview
First you have to create an API in your Google account. This will enable other programs to connect to your data (provided they pass the correct credentials). After that you have to adjust the .mez-file to include the client id and client secret that identify your API. This data will be generated when setting up the API. After you’ve adjusted the .mez you have to store it in a dedicated folder so that Power BI can access it. Lastly, if you want to use this connector in the Power BI service, you have to add the .mez-file to a gateway.
Setup Google API
Go to the Google Developer API and if you don’t have a project yet, just create one:
NEW: A OAuth consent screen will pop up and you have to give your app a name and fill in “PowerBI.com” into the “Authorized Domains”:
Next go to “Credentials” -> Create credentials and choose “OAuth client ID”:
Choose “Web application”, adjust the “Name” if you like and paste the redirect-url into “Authorized redirect URLs”: https://preview.powerbi.com/views/oauthredirect.html
This will return the client ID and secret for your connector: