These instructions are written for FreePBX 10.13.66 with SIP 5.4.0 and tested with Polycom VVX 410 phones. They will probably work for other systems and Polycom phones as well.

By default, when a Polycom phone is paged it rings once before auto-answering. FreePBX also plays a beep by default before beginning the page. We only need one of these sounds to play. It is relatively easy to turn off the beep in the FreePBX settings. Disabling the ring is more difficult. If you have purchased the EndPoint Manager module, it can be done by editing the SIP-Interop.cfg file. Look for the se.rt.ringAutoAnswer section.

<se.rt.ringAutoAnswer
 se.rt.ringAutoAnswer.callWait="callWaiting"
 se.rt.ringAutoAnswer.micMute="0" 
 se.rt.ringAutoAnswer.name="Ring Auto Answer"
 se.rt.ringAutoAnswer.ringer="ringer2"
 se.rt.ringAutoAnswer.timeout="2000"
 se.rt.ringAutoAnswer.type="ring-answer"
 >
</se.rt.ringAutoAnswer>

Delete the entries for se.rt.ringAutoAnswer.ringer and se.rt.ringAutoAnswer.timeout. Change the value for se.rt.ringAutoAnswer.type to "answer". The config section should then look as follows.

<se.rt.ringAutoAnswer
  se.rt.ringAutoAnswer.type="answer"
  se.rt.ringAutoAnswer.callWait="callWaiting"
  se.rt.ringAutoAnswer.micMute="0" 
  se.rt.ringAutoAnswer.name="Ring Auto Answer"
  >
</se.rt.ringAutoAnswer>

Rebuild the configs and reboot the phones for all your paging pleasure.

I have submitted a feature request to FreePBX to have them include an easy GUI config for this in the future.