One of the 4 toilets in the house runs water through the fill tube even though it has not been flushed. This happens when a large flow of water from another water user such as filling the washing machine or flushing one of other toilets. Once the large flow ends the water flow from the suspect toilet stops. The suspect toilet does not flow water when none of the other large water flow sources are flowing.
Flow through the fill tube when there is high water draw from some other water user
toilet
Best Answer
There are several types (and many brands and models) of toilet fill valves, aka ballcock valves.
The type most susceptible to pressure fluctuations are those that utilize a "weir diaphragm" assembly because these rely on inlet pressure to keep the valve closed. An old stiff (or worn/soft/contaminated) diaphragm can exacerbate this susceptibility such that a pressure bobble can cause the valve to open.
You should open the valve to inspect the bonnet area and diaphragm for problems; cleaning/flushing all parts and perhaps replacing the diaphragm (replacements are commonly available, specific to each brand and model).
Another option would be to just replace the entire valve.