Is it permissible to participate in a simple birthday celebration confined to family members only, and what is the ruling on refraining from participating in this case?

Celebrating a birthday is an Eid (festival) to which the rulings of Eids apply. An Eid is defined as a recurring gathering (annual, monthly, weekly) that involves specific acts. A Muslim must avoid places of disobedience if they cannot change the wrong. If they can change the wrong with their tongue, then they may be present, based on the Almighty's saying: "So do not sit, after the reminder, with the wrongdoing people And there is no accountability upon those who fear Allah concerning them, but only a reminder, that they may fear Allah." So, one must explain the ruling of celebrating these Eids to their relatives with kindness. If they insist on the wrong, one should part ways with them.