This is a crude drawing of a circuit could could work for your application.
View attachment 188054
It uses a SSR (solid state relay) that handles the AC load of the alarm device. This also assumes that the alarm is an AC line voltage device. The camera relay is also assumed to be a NO (normally open) device and only completes the circuit when it is triggered.
The 9 volt DC battery would be a temporary testing source for camera relay power. However, if the camera outputs 12 VDC at a max load of 30 ma, then the camera can be directly connected to the SSR without the battery power source.
To check, use your DMM or DVM to see if there is any DC voltage on those two output connections when triggered. Some cameras may provide an output voltage while others may provide a dry relay contact.