There are several plugins that can help you talk to a API like
MBS Plugin - Monkeybread Software - MBS FileMaker Plugin 7.1
ok, i used Often this BaseElements plugin.
But my problems is:
At the precise moment when the call comes in, the switchboard must call Filemaker (via API) and the operator must then accept the call by displaying all the data related to the number you are calling. If the number exists in the archives of Filemaker.
Then it is up to you to build a fast and reliable solution that does what you arw asking for. The API can trigger FM script that search in your Phone Number field (that should be indexed) and the result should be displayed