The floor drain would have a p-trap to prevent sewer gas.
It would need to be vented, and the vent can tie back into the washer vent, at 6" above the flood level of the fixture.
The floor drain would need a trap primer to keep it wet.
Two ways to do this are"
A dedicated trap primer connected to the water supply.
A tee on the standpipe of the washer that diverts water to the p-trap. Everytime you run a load, a bit of water would wind up at the p-trap from the water being discharged.