How to identify the positive and negative connectors on a PCB connector
The connector and the PCB itself gives some hints:

The frontside of the IR-PCB(lower left in the picture) has a lot of LEDs exposed and one optically shielded from the other(down in the middle). This is an indicator that there is some measurement happening on a single IR LED and thus there is at least one wire carrying this signal.
Another hint is the circuitry one the backside(lower right in the picture) of the IR-PCB. Down in the middle are some SMD components which look suspiciously like a switching regulator(one big coil, a diode, an IC and some small components)
The connector on the frontside of the IR-PCB gives some additional information. First in most cases its common to put ground on Pin 1 of a connector. Pin 1 is indicated by a special solder pad(in this case square) or other special silkscreen markings(like the triangle next to &quot;CN1&quot; on the backside). In this case I'm not quite sure because we can clearly see a thick trace emerging from pin 1, which is uncommon because mostly ground is a solid copper pour all over the PCB. This could be a power trace.
Looking further at the connector solder joints we can see another smaller trace emerging from pin 4 down under the LEDs. This could be some kind of signal. Pin 2 and 3 seem to be joined together and no clearly visible trace is emerging from them. This is an indicator for a ground connection, they could be connected to the copper pour.

So if these conclsions are right we got the following pinout:
Pin 1: Power
Pin 2: GND/EN
Pin 3: GND/EN
Pin 4: SIG (Look on edit)
From a EMC standpoint its quite a good idea to have (analog) signal cable spatially seperated from a power line on which one end a switching power supply draws current and produces noise. Last but not least we can see a capacitor on pin 1 of the connector on the green PCB, which also indicates that this is a power line.
For further investigation I would suggest that you measure the traces with a multimeter. Maybe measure the resistance between pin 2 and 3, if its low the pins are connected. Then measure the resistance between the power socker on the camera and the pins on the connector. This should finally give you the right wiring information.
No guarantuee that you can put 12V on the IR PCB, maybe its stepped down one the green PCB or elsewhere.
Hope my guesses and analysis of your PCBs could help.
Edit: As Passerby suggested there could be an enable pin for the converter. So maybe just Pin 2 or 3 are connected to ground and the other one is for enabling the converter. This means that you may need an additional signal to turn on the LEDs.