Our house is about 18 years old. Recently one of the toilets has started making a moaning sound which seems to start after the tank has refilled after flushing. Is this something which we can fix ourselves, or should we call a plumber?

Yes! This is a home owner repair. A few questions first. Does the noise start while the tank's filling or a while after it's filled.
When the tanks's filled how far beneath the white overflow tube is the water level? Have you tried adjusting the flow at the angle stop to see if that stops the noise?
Back to you, Tom