Track loader actuators: Provide reverse capability for two DC motors, on board snow blower attachment
You are on the right track (no pun intended) with two double-pole-single-throw relays, but the wiring is not optimal. Your sketch feeds the motor current through the joystick buttons; when you release a button there will probably be an arc that eventually damages the switch contact.
Instead, keep the relay coil/button circuits separate from the relay contact/motor circuits.

Hot feed C3 -&gt; Joystick button D -&gt; coil of C3 -&gt; ground.
Hot feed C2 -&gt; Joystick button C -&gt; coil of C2 -&gt; ground.
Hot feed C3 -&gt; lower contact of relay C3 -&gt; M1 brown.
Hot feed C3 -&gt; upper contact of relay C2 -&gt; M1 green.
Ground -&gt; upper contact of relay C3 -&gt; M1 green.
Ground -&gt; lower contact of relay C2 -&gt; M1 brown.

Only use this if the joystick construction ensures button C and D are never pressed at the same time. If that happened, both relays operate at once, shorting out the battery and hopefully just blowing a fuse but possibly burning up wires.
With additional information (from OP's comment) we find there are already relays in the track loader, operated from each button. In this case, first check to see if the track-loader's outputs are grounded when inactivated. That could be the case, because it would help provide braking for electric motors when the joystick is release. If so, just connect the output of C to motor brown, output of D to motor green and you're done!
If the relays are only SPST, then it should be a simple matter to just replace these relays with SPDT units of similar size and configuration. Wire them with the COM contact as the output, the NC to ground, and the NO to fused hot. Now, as above, just connect one relay's output to motor Brown wire, the other relay's output to motor Green. Done!