What is the ruling on going for Umrah with one's mother and brother for a birthday, knowing that I do not celebrate this innovation?

It is not permissible to celebrate a person's birthday, as it thereby becomes an 'Eid (festival), and it is a custom introduced by non-Muslims. This falls under the blameworthy imitation forbidden by Islamic law, as the Prophet (peace be upon him) said: "Whoever imitates a people is one of them."

Muslims have no festivals other than Eid al-Fitr and Eid al-Adha, as the Prophet (peace be upon him) said: "Indeed, Allah has replaced them for you with what is better than them: the Day of Adha and the Day of Fitr." If the celebration is performed as an act of worship, it is a rejected innovation, as the Prophet (peace be upon him) said: "Whoever introduces into this matter of ours that which is not from it, it is rejected."

There is no harm in performing 'Umrah on one's birthday without intending it as a celebration, as long as there is no intention of a specific act of worship solely for that day.