Is it permissible to ask one's mother about an important matter—which appears to be negative—that happened to one's cousin (son of the maternal aunt), and would that be considered backbiting (gheebah)?

Backbiting is to mention your Muslim brother with what he dislikes of physical or moral attributes, or anything related to him, whether verbally, in writing, or by संकेत. This includes the body, religion, worldly affairs, self, physical appearance, character, wealth, family, and so forth. Its defining characteristic is: mentioning him with what he dislikes. If your question about your cousin leads to backbiting, then it is forbidden, except in exceptional cases necessitated by a legitimate Islamic interest, such as assisting in changing a reprehensible act, like mentioning the person to someone who can denounce it if you are unable to do so. And whoever knows the prohibition of backbiting will hold their tongue and refrain from asking about what does not concern them; because speech may lead to what is unlawful.