Is a daughter's communication with her close friend, whom her mother has forbidden her to communicate with, considered disobedience if it is without the mother's knowledge?

We do not know the motive behind your mother preventing you from communicating with your friend. If the motive is legitimate, such as the girl being bad and your mother fearing corruption for you, then it is your duty to obey your mother, and your disobedience to her is a sin and an act of disrespect. If the girl is righteous, try to convince your mother to allow you to communicate with her, and seek help from those she listens to. If she agrees, then praise be to Allah. Otherwise, communicate with your friend without your mother's knowledge, because obedience to parents is obligatory in matters that benefit them and do not cause hardship for the child.