How to make a Jenkins API call when using "Log in with Google" as security?
Typically you authenticate API calls (I assume you mean HTTP REST API; there is also a Groovy API) using an API token.  The API token can be generated from a user's profile page while logged in as that user (Profile page > Configure > API Token).  More information about generating and using API tokens is available in the official documentation.  There is also a question about API tokens posted to StackOverflow.