The noise may be caused by incorrect meshing of the gears. Too little backlash will cause noise and premature wear. Too much backlash will cause noise and an inconsistent feed rate.
The gears should be meshed with a piece of card or paper between them, about 0.003" (0.075 mm ) thick to set the correct amount of backlash. Obviously, remove once the meshing has been set.
You must expect some noise. Even precision ground gears emit some noise.
Heavy grease will reduce noise, but as you say, should be kept away from belts and pulleys.
The gears on your lathe are unlikely to be as precisely manufactured as those in your car gearbox, and they make some noise.
Howard
Edited By Howard Lewis on 22/08/2020 12:22:20