Wiring thermostat fan ON
After looking at all the photos, it looks like the fan is controlled by a fan and limit switch (the black thing with a silver dial and three black wires). 

So basically, the thermostat calls for heat by energizing the TH W terminal on the gas and ignition controller (Black Honeywell box with all the wires connected to it).  This opens the pilot, turns on the spark, and ignites the main burners. Once the temperature in the heat exchanger gets up to the FAN ON temp., set on the fan and limit switch, the blower comes on. When the thermostat is satisfied, it removes power from the TH W terminal. This tells the ignition and gas controller to turn off the main burner.  When the temperature in the heat exchanger falls below the FAN OFF temperature setting, the blower turns off.

To control the fan via the G wire from the thermostat, you're going to have to use a relay to turn on the blower.  You'll energize the coil of the relay using the G wire from the thermostat, and the 24V common terminal from the transformer.  You'll have line voltage (120V or 240V) connected to the common terminal on the relay, and a wire to the blower connected to the normally open (NO) terminal on the relay.  

If you don't understand the information provided in this answer, or don't feel comfortable making this type of modification to the furnace. Please don't hesitate to contact a local licensed HVAC technician.

Side Note:  The gas and ignition controller is installed upside down