Installing a relay to control duct fans!
Your G terminal isn't meant for that
On gas furnaces, the G terminal is basically an &quot;override&quot; input to allow the end user to turn the fan on without a call for heat or cooling.  As a result, your results with your current fan relay aren't totally surprising; in fact, it shouldn't turn on at all unless you set the fan switch on your thermostat to ON.
But, your goal is still achievable
However, you can still get where you want to go.  You'll simply need a different relay with a 120VAC capable coil (a RIBU1C mounted to a junction box will work just fine for this) instead of your current, 24VAC-coiled fan relay.  The coil of said relay connects to the EAC-H and EAC-N terminals on your furnace's control board using crimp-on female quick-disconnect terminals, and the contacts switch the fan on and off just like you are doing with your existing relay.  Note that you'll need to use mains-rated wiring methods for the connection from the furnace to this relay; a piece of 14-2 AC or MC that uses stranded wire is your best bet, or you can use 3 14AWG THHNs (or 2, in a metal conduit) in your choice of conduit flavors if that's your preference.