I've searched the web and didn't find anything related to the availability of Q Enterprise Messenger and the possibility to retrieve the messages once they have been deleted from the device/PC. Also, this Messenger doesn't appear to be available anymore. Instead, the AT&T company released another application available for modern devices.
You can check it on the following page, but for your inquiry there isn't a viable solution anymore: https://www.att.com/shop/apps/att-messages.html
