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 (definition); otherwise, it is not. The mahram (definition) 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 (definition): "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."