It seems that
"make noise: emit a noise (could be any noise such as one from music, TV, people talking, ...)" Source
"make a noise (about something): (informal) to complain loudly (talking about human only)" Source
"make noises (about something) (informal):
The company has been making noises about closing several factories.
But I am not sure if my research is right or not?
Noise can be countable (refering to individual sounds that are noises) or non-countable (refering to a stream of sound with qualities like noise). So make a noise does not automatically mean "complain" and make noise can mean this too.
Make [a/some] noise about X can be figuratively used to mean "doing something to gain attention about X" but I'm not sure that association is strong enough to really be an "official" well-known saying, but certainly a strong implication.
I would just like to add that with children you might say "make a noise" "make a sound" "don't make a sound". What noise does a dog make? What does a cow say?"
Or in general, my refrigerator is making weird noises. It's making a whistling sound.
Make noise to complain, yes I've seen it. But I don't hear it in common speech very often. Oh, the neighbors are making a lot of noise. (I would interpret that first as they are having a party and making a noise disturbance instead of them "complaining about something". )
