Settings > Apps > Google Play Services > Notification settings > off

Settings > Apps > Google Play Store > Notification settings > off

That's it.

Updated: Jun 27 2019:

The new google x86 emulator with play services is exactly like a device where you can install apps from play store and all other things.

Previous Answer:

I faced similar problem while trying google maps in emulator. While some people have been able to run google play services in their emulator by downloading google play services application from their mobile, making the apk file and then installing that file in emulator, it is really complicated process and works for certain api level only. Same is the case with emulator using google api's that shift with builtin maps and google play services. The maps application will work in emulator while your application asks for update to google play services. So the wise thing to do is to test it in real hardware.

I had this same problem. I opened my emulator, went to Settings -> Apps -> Google Play services, and noted the version (in my case, 8.1.65). Then I opened my build.gradle and saw that the version there for com.google.android.gms:play-services was 8.3.0. I changed it to 8.1.0 and it worked on the emulator.

Navigate to settings--> apps in your emulator and then find Google Play Services, check the version number and use it in you build.gradle

Hope this helps

Any device or emulator will need to have at minimum the version of Google Play Services that your app is compiled with. Standard practice is to prompt the user to upgrade if the version of Google Play Services on the device is lower than needed.

Here is the new way to prompt the user to upgrade Google Play Services. Call this checkPlayServices() method when your app is first launched:

public static final int PLAY_SERVICES_RESOLUTION_REQUEST = 999;
GoogleApiAvailability googleAPI;
private boolean checkPlayServices() {
    googleAPI = GoogleApiAvailability.getInstance();
    int result = googleAPI.isGooglePlayServicesAvailable(this);
    if(result != ConnectionResult.SUCCESS) {
        if(googleAPI.isUserResolvableError(result)) {
            googleAPI.getErrorDialog(this, result,
                    PLAY_SERVICES_RESOLUTION_REQUEST).show();
        }

        return false;
    }

    return true;
}

If the device has an older version of Google Play Services, the update prompt will be shown: