Is it permissible for you to be a mahram to your married paternal cousin, who is 50 years old, during Hajj or 'Umrah?

If you are a foster brother to your paternal cousin, then it is permissible for you to travel with her for Hajj or Umrah; otherwise, it is not. The mahram is the husband, or someone to whom the woman is permanently forbidden due to kinship (like the father, son, brother, paternal uncle, and maternal uncle), or due to a permissible reason (like the mother's husband, the husband's son, his father, and her foster brother); based on the hadith: "It is not lawful for a woman who believes in Allah and the Last Day to travel for a journey of three days or more, unless her father, or her son, or her husband, or a dhul-mahram is with her."