I'm not big on alarms and usually have to do a bit of reading, but it looks like it should work. The outputs to the relays... do you have more information about them?
If outputs 1 and 2 are sources, and they are pulled high to say 12v when they are active, then you might be advised to include some flyback protection diodes across the relay coils to avoid potential damage to the electronics when they are de-energised and the magnetic field collapses. This is particularly important if the outputs are transistor based.
If outputs 1 and 2 are sinks (i.e. they are effectively open circuit, but get connected to 0v when active - typically via an open collector transistor), then you'll need to connect the common for the relays to +12v and the same advice applies regarding flyback protection diodes.
The inputs to the gate motor... are they momentary inputs (i.e. pulse it to trigger open/close) or do they need to be on all the time? If it's the latter, you could potentially get away with a single relay operating in a changeover configuration. I might be inclined to do this, as connecting both the Open/Close of the motor could have some undesirable effects (can't say for certain). In this configuration, I would have it so that you energise the relay to close the gates so if the alarm/control/relay fails and the contacts open, the gates will open as well.
The other point about this is, what triggers the gates to close again? If you're not careful, you could end up in a situation where the PIR triggers them, the gates open, then the PIR deactivates and the gates close. If the vehicle or whatever is still there, damage to vehicles or people could be the end result.
The electric gates I've seen/useds all have multiple sensors, including infrared beams and induction sensors in the road to detect vehicles and manage the gate states accordingly. In my opinion, you may be better to use a gate controller and trigger than with the alarm rather than try and manage the open/close states of the gates yourself.
