On most intercom systems such as the one you have indicated the switch will most likely be a dry N/O contact and as McBryce has pointed out you might be able to put your relay contact in parallel with it, I cant see any wires for the release going into those terminals so there might be a bit more to it.
Yours appears to be a basic system but some systems require the handset to be off hook or even in communication with the gate station before that button is active, other variations will have the control contact in the gate or front door station itself and this is more common, you will need to test this button before you go any further to confirm the units operation, also find out the make and model of the intercom to download any relevant documentation.
Most boom and gate controllers have an input that requires a momentay closure on these terminals thats what the intercom button does and if it's over a long distance then it may need a second or two to activate, now this has been discussed before on the forum and there are a number of things to consider and these are important, if your MCU or Pi loses power then you need to make sure that the relay does not trigger or bounce on a restart or for that matter any other unexpected event.
Also some boom and gate controller systems will incorporate a ground sensing loop or photo electric beams to ensure that the gates or boom will not close whilst there is an obstruction or vehicle present, some may have a sense loop on the entry side so that the boom or gates cant be activated unless a vehicle is waiting to get in, if these protections are not incorporated in the system then your addition or modification could put people or property at risk, not all systems have these loops or beeams.
Additionaly, anything you add or modify needs to be clearly documented and displayed so that other technicians in the future know exactly what you have done or altered and relevant permissions must be sought before commencing any works.
All else aside, please keep us updated and good luck.
Muttley