Is a daughter, married and having her own car, obliged to obey her mother if the mother forbids her from driving to visit her, after having previously encouraged her to do so, in order to avoid problems with the daughter's brother?

Obedience to the mother is obligatory in matters that are beneficial to her, as long as no harm befalls the child. This is like keeping the mother away from problems with the questioner's brother. One should seek another way to visit relatives, such as her coming with her husband, or her brother, or other mahrams, so that Allah may place goodness in her dutifulness and the pleasure of the Merciful Lord, for His pleasure is in the pleasure of the parent.