You have a VoIP phone system running the FreePBX distribution of Asterisk with Digium phones connected to it. Your client says "I want interoffice phone calls to be intercom style - no ringing". Here's how to accomplish that.
First, enable it on a per-extension basis in FreePBX.
Second, install Webmin and edit the FreePBX database to change the intercom signal from "Ring-Answer" to "Intercom" (to eliminate a preliminary ring and only have a "beep" sound).