Quran for Android does not share any user identifiable information with third party entities. Quran for Android does use Google’s Fabric service for anonymously tracking analytics and crashes, but all of this data is not tied back to a certain person.
Quran for Android used to request
READ_PHONE_STATE permission to stop recitation if a phone call comes in during an audio playback. That was removed on September 27, 2015 here. People running versions released prior to that day may still have a version that requires this permission.