Drupal way of sending push notification to android devices
Try the Push Notifications module. I've never done this -- but you're basically asking how do I send X message type to a user (SMS, Email, whatever). I've sent emails and SMS messages before ...

Beyond wanting to send this notification from within drupal/php ... there isn't anything really drupal specific in your question in terms of the drupal API. 

If you dont userstand the protocol or APIs you want to talk to I would suggest you research them more as well. I would start by looking at the source code for the PushNotifications module and see how it handles the logic of what you are asking about ...