Why did this app get removed from the Play Store? What was this app for?
Most likely the software has been discontinued and because of that, the developer removed it from the Play Store.
I've used all the Android market websites to look for this application and it appears that it's gone missing.
As an alternative, use Helium since I've used it and it works good: https://play.google.com/store/apps/details?id=com.koushikdutta.backup
Q Backup Agent was an application used to create backups of your phone.