How do you format the set_body option for WP_Rest_Request?

WP_Rest_Request is not a way to make outgoing calls to remote REST APIs, it’s a data object core uses to pass around information about an incoming request. It’s basically something you recieved, it isn’t something you can send.

If you poke your sites REST API with a request, WordPress creates and populates a WP_Rest_Request object and uses it to help handle the request.

So you cannot use this class to make requests to Google.

For that, you want to use the WP_HTTP APIs, such as wp_remote_get, or wp_remote_post.