When I click the Google sign-in button, it shows the following dialog.

The emulator does not support the Google Play Services. That is why it's throwing an error. To develop using the APIs, you need to provide a development device such as an Android phone or tablet.
Google Play services out of date.Requires 4132500 but found 4033530
To get the play services in your emulator, you have to install the APK file on your own.
Here are the APK gms and vending files.
Install both the APK files using the below command.
adb install com.android.vending-20130716.apk
adb install com.google.android.gms-20130716.apk
And if it's already installed then to upgrade an existing emulator, you need to uninstall the previous versions using the below command and then install it again:
adb uninstall com.android.vending
adb uninstall com.google.android.gms