Even though a toilet only uses cold water i have seen some toilets do this.
Tankless water heater noisy pipes.
If you hear a weird groaning sound from the pipes or the tankless hot water heater when you flush the toilet that could be the culprit.
Another standard issue that causes loud noises in tankless water heaters is a dirty flow sensor.
If the running water noise continues once the tank is filled you may want to call a professional plumber to investigate the situation for you.
A blocked or improper ventilation and even a defective or wrongfully set pressure valve can also make your tankless water heater loud.
Generally speaking there is a lower chance of hearing strange noises with a tankless water heater vs tank model.
If your water heater is making noise like water is running you probably have a leak or a broken pipe.
Although keep in mind that it is normal to hear water entering the tank when hot water is being drawn.
Others reasons for a noisy tankless water heater model may be an issue with the burner the fan may be dirty or possibly a leak in the sealed combustion resulting in irregular gas combustion.
Or you can try to run down the leak yourself by checking the following.
The sensor controls the amount of gas sent to the unit.
Water vapor is a byproduct of combustion although in this case the vapor contains elements such as methane nitrogen and oxygen.
In that case you need to have a plumber install a check valve in the water line.
Inside a condensing tankless water heater the combustion process superheats the water that enters your shower sink etc.