Toilet noise

Yes — most likely it is the toilet fill valve / ball valve inside the cistern vibrating as the cistern refills. A fog-horn or humming noise after flushing is commonly caused by debris, limescale, or a worn diaphragm/washer in the fill valve. The fact that the noise reduces when you open the cold tap strongly suggests a pressure/resonance issue on that cold-water supply as well.

What I would suspect first:

worn or dirty fill valve
partially restricted service valve / isolation valve
high water pressure causing resonance
less commonly, pipe vibration / water hammer in the cold feed.

A simple check:

Take the cistern lid off.
Flush the toilet.
When the noise starts, see if it is coming from the fill valve area.
If you gently lift the float and the noise stops, that points strongly to the fill valve rather than the pan or waste side. This is a common diagnostic approach for this type of fault.

Best fix:

first try cleaning the fill valve and checking for debris/limescale
if that does not solve it, replace the fill valve
if the problem continues after that, have the incoming pressure and any service valve checked.

The "foghorn" sound you hear after flushing is most likely caused by a faulty toilet fill valve also known as a ballcock valve

That foghorn sound is usually caused by a vibrating or worn diaphragm washer in the fill/ball valve inside the toilet cistern. When the cistern refills after flushing, the water flow makes the faulty washer vibrate, producing the loud humming or foghorn noise. Turning on the cold tap reduces the pressure in the pipe temporarily, which is why the noise gets quieter. The fix is to replace the fill/ball valve or the diaphragm washer inside it, which is a quick and inexpensive repair. Once replaced, the noise should stop completely.

That foghorn-type noise is a fairly common issue and usually points to a faulty or vibrating fill valve inside the toilet cistern. Over time, the internal diaphragm can wear out or harden, causing it to resonate when the cistern refills — that’s the noise you’re hearing. The reason it quiets down when you turn on the cold tap is because it changes the water pressure slightly, which temporarily stops the vibration.
The fix is straightforward — replacing the fill valve or the diaphragm washer inside it normally solves the problem completely. It’s a quick and inexpensive job for a plumber.
— Fixzer Plumbing

The wc inlet valve diagram washer commonly the ballvalve type (arm with a float).
It can also be from a ceramic disc internal on a cold tap nearest the toilet.

Water hammer is the term used for this noise

There is a loose fitting or worn out washer in the cistern intake valve that is allowing air to enter the system when water is demanded by the toilet.

The full valve needs replacement

Could be from the filling valve being faulty or dirty.
Could also be a clog in the cold feed sending water to the inlet valve.

Water pressure

This is often caused by a worn or partially blocked fill valve or diaphragm washer in the cistern. When water rushes through a faulty or loose washer it can vibrate and create a “fog horn” sound.
Turning on the cold tap changes the pressure, which is why the sound eases.

1. Water Hammer / Pipe Vibration
• When the toilet fills quickly, the sudden stop of water can make the pipes vibrate, producing a loud, low-pitched noise.
• The fact that opening another tap (cold water) reduces it suggests it’s related to sudden water pressure changes in the supply pipe.

2. Faulty or Worn Fill Valve
• Inside the cistern, the fill valve (ballcock or modern valve) controls how fast water enters.
• If it’s worn, partially blocked, or faulty, water may rush in at high speed, causing vibration/noise.
• Older toilets with metal valves are particularly prone to making a “fog horn” or “whistling” sound.

3. Pipe Resonance / Loose Pipes
• If the water supply pipes aren’t properly secured, they can resonate when the toilet fills.
• The noise may be reduced if another tap is opened because the water pressure is shared, lessening the pipe vibration.

The shut off valve that feeds the toilet may be faulty. The valve shut off screw may not be in the vertical position, which would cause undue pressure when the cistern is being filled. The syphon
might be faulty

New toilet cistern inlet valve is required

The noise is your toilet tank refilling, turning the cold water tap on will reduce the cold water pipe pressure, it makes less noise when toilet water tank is refilling due to water pressure drop.