To act like a spoiled child or to 'coo' at someone by making a nasal sound (an idiomatic expression for behaving dependently).
子猫が飼い主に鼻を鳴らして甘えている。
The kitten is cooing at its owner, acting affectionately.
彼はいつも鼻を鳴らして母親に甘えている。
He always acts spoiled and coos at his mother.
To make a nasal sound such as snorting or sniffing (the sound of the nose; can indicate annoyance, amusement, or sniffing).
彼は不満そうに鼻を鳴らした。
He snorted in displeasure.
寒さで彼女は鼻を鳴らしながら歩いていた。
She walked along sniffing because of the cold.