Is the belief of a relative of mine that covering a woman's hair is not an obligation but a custom, and that what matters is a woman's modesty, considered apostasy that expels one from the faith?

Saying that covering a woman's hair is not obligatory is ignorance and misguidance, for a woman's hair is awrah (private parts that must be covered) by scholarly consensus, and it is not permissible for her to expose it in front of non-mahram (definition) men. Your relative should fear Allah and stop speaking about Allah without knowledge, for permitting what Allah has forbidden is a grave matter. This belief does not constitute apostasy that expels one from the faith, because the obligation of covering a woman's hair has not reached the degree of being "known by necessity from the religion" in this era due to the widespread ignorance and tabarruj (immodest display). However, it is a dangerous statement that could lead to something more terrible.