My wife and I recently moved into a new house and during the day and night we hear what sounds like the toilet cistern either partly filling or flushing. This happens to BOTH bathrooms, more than once and at irregular times.
What can I do about this??

It sounds like there is a leak at the bottom of the tank.
When the water level in the reservoir goes down, the float drops and the valve opens to fill the tank
Try putting some dye in the back of the toilet and see if it shows up in the bowl.
If the dye shows up in the bowl then the flapper seal is not holding and will have to be replaced.
This is usually the problem.