Is the questioner considered sinful for repelling injustice from his mother, which led to a quarrel and his brother leaving the house, and people circulating talk about the matter, knowing that he cannot ask for forgiveness?

If the matter is limited to merely responding to slanders and preventing them without aggression or overstepping boundaries, then there is no sin upon the doer, and they are not required to repent or seek forgiveness. This is based on the Prophet's (peace be upon him) saying: "Help your brother whether he is an oppressor or oppressed." And he explained helping the oppressor as preventing him from committing oppression. It is also advised to strive for reconciliation between Muslims, based on Allah’s (the Most High) saying: "The believers are nothing else than brothers. So make reconciliation between your two brothers," and the Prophet's (peace be upon him) saying: "Shall I not inform you of something better than the degree of fasting, prayer, and charity?" They said: "Yes." He said: "Reconciling disputes, for indeed, discord among people is the razor [that shaves away good deeds]." Repentance is obligatory for overstepping boundaries, and if the matter involves the right of a created being, then one must seek their forgiveness.