From what you have explained in the question, I guess you are talking about cross client resource access using Google sign in. And specifically you seem to be interested in obtaining the Id token once and use it without having to obtain on each subsequent API call.
This more or less is synonymous with the offline access mechanism.
In offline access, the Client I.e. the Android app asks for user authorisation for requested scopes. Upon authorisation, instead of issuing an access token, auth server returns a short lived authorisation code which can be used to generate an access token and refresh token.
The client then can pass the authorisation code to the backend over a secure connection. Backend server can retrieve the author token and refresh token and store them in a secure location. The access token is short lived and can be used to access scoped resources for a short time and refreshed from time to time using the refresh token. The refresh token does not expire but can be revoked. If revoked, server app should ask the client app to re-fetch the author code.
Please go through this link which details the complete infrastructure along with the steps to be followed both by client and server app -
https://developers.google.com/identity/protocols/CrossClientAuth
Now coming to your question, you should use a slightly different API to obtain the auth code. Check out this API -
https://developers.google.com/android/reference/com/google/android/gms/auth/api/signin/GoogleSignInOptions.Builder.html#requestServerAuthCode(java.lang.String)
Sample code at - https://developers.google.com/identity/sign-in/android/offline-access