That this can work, you must obviously be ready to handle (at least) redirects caused by the API calls and emulate a browser sufficiently that the "external Google page"'s expectations are satisfied. If you can satisfy these conditions, then there should be no other big problems.
If your "restapi" calls are part of normal user interaction inside the browser, then a previous login in the standard browser way should be sufficient to authorize the API calls.
If your "restapi" calls are in a special context, you might be able to use a special user. E.g., I often use a Zope "Manager" (authenticated outside Plone) for such tasks.