Can I run a C wire from Fan Relay

We’ve just moved into a new home with an heat only, oil, forced hot air furnace. The current thermostats are very old and we’d like to update them to something like a nest. The furnace is currently wired to the thermostat via two wires (T T) and there is no common coming from the terminal to which the thermostat is connected.

The nest says you can run it with a two wire system, but I don’t want the nest pulsing the furnace to recharge its battery all summer when the system isn’t in use and so would prefer to power it via a common wire.

My current plan is to buy a transformer and wire that to the C and Rc terminals on the nest. BUT, I wanted to look into another option.

There is a box on my furnace that does have a terminal labeled C. I believe it is a fan relay. Currently there is something running to that C wire, but it disappears into the furnace (pictures of the furnace system is attached). Could I buy some 18/3 wire, run two of those wires from the T/T terminals and then connect up a second wire to the C on the fan relay? Something tells me that’s a mistake and I should go with my first plan, but if this second plan works it seems like a cleaner solution.