Whenever the unread count changes in Textra SMS, it broadcasts an intent with the action set to "com.textra.UPDATE_UNREAD". The intent will have an integer extra called "count" with the unread count.

You can receive the broadcasts in the normal way, by either registering a receiver in the manifest of your app, or by programmatically registering a receiver. 

The official documentation is here: https://developer.android.com/reference/android/content/BroadcastReceiver.html and a succinct tutorial is here: http://www.vogella.com/tutorials/AndroidBroadcastReceiver/article.html#broadcastreceiver

Any queries please contact developer@textra.me.

Note: This broadcast is provided only from Textra SMS v3.22 onwards.