Most commonly the ice maker needs to be reset or replaced. There are a few other variables like cabinet wiring and water line issues which are pretty rare.
Usually I have the customer turn the unit off and back on then add water to the ice mold ( if it's empty ) and see if it cycles and how many times it drops; if it does cycle. If the ice mold is full and frozen I do a power reset and then a test mode cycle.
TEST MODE: Make sure the ice maker power ( rocker switch in "I" position ) is on. If the green cube size leds are lit on the face of the ice maker push and hold the ice cube size button until the ice maker begins a test cycle ( 3-10 sec ). The row of fingers above the mold ( where the water is held and turned into cubes ) began to rotate. At the end of the cycle, *when the fingers come back near their original position, it should fill with water.
( * Sometimes an IM will fail leaving the fingers in a position other than where they should be at the end of the cycle. On this type IM the fingers are supposed to rest slightly above the edge of the ice mold portion of the IM body on the side facing you ).
If the switch is on but the lights are not lit the IM is dead.
If it cycles but does not fill, move on to checking for 115v at the water valve when test cycling the IM again. If no voltage is ever present suspect the IM. If voltage is present make sure water is flowing through the valve by disconnecting the output side water line when testing. Voltage present but no water flow = bad water valve.
If no voltage is present at the water valve and the IM leds are lit: The water valve relay is on the M.PCB but is digitally told to actuate by the IM pcb. The water valve terminal at connector 3 / pin 3 and a common should read 115v when the IM is at the fill position. If 115v is not present either the M.PCB or the IM is at fault. Since the IM communicates with M.PCB digitally there is no way to test this communication with basic tools. At this point it becomes an educated guess. Ice maker failures are very common on LG refrigerators and would be my choice to replace.