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Google Play services out of date.Requires 4132500 but found 4033530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
I found the solution on the page How to install Google Maps v2 in Android Emulator.
It only works with Android 4.22, but I think that the same idea can be applied to version 4.4 with the appropriate files.
Download the three files listed on the previous page: com.android.vending-19032013.apk, com.google.android.gms-19032013.apk and Google play services lib r5.
Create an AVD device with a normal target (Android 4.22, API 17), with Intel Atom (x86) if you want to use.
Install the two APK files from Android console.
Create your app for Android 4.22
DO NOT USE the google-play-services-lib of your system. You must use the third downloaded file.