Are you certain that your existing circuit is NC?
The typical garage door opener button is a momentary switch. Many third party remote devices, including two that I have installed, tap directly into this circuit. Because the contacts short across the existing switch, activating one, two or three is insignificant. These switches are connected in parallel.
It would be a good idea to use an external isolation circuit to the Arduino, either a relay or SSR in order to ensure that there is no "injected" power from your circuit to the garage door circuit.
If your circuit is indeed an NC type, you would connect your new relay or SSR in series, using the device's NC connections. The locations for tapping into the circuit would be where your existing button resides or where those two wires engage your garage door mechanism.
I suppose my older garage door opener is NO, while newer models would use a different architecture, aka NC.