Is a single girl's bearing the expenses of her grandmother's (her father's mother) Hajj considered charity, or is she obligated to spend on her?

Your act of paying for your grandmother's Hajj expenses falls under cooperating in righteousness and piety, and showing kindness to grandparents, as Allah Almighty says: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression). For this, you will have the increased reward of charity and maintaining kinship ties, for the Prophet, peace and blessings be upon him, said: "Charity given to the poor is a single charity, but given to a relative, it is two: charity and maintaining kinship ties." In any case, you are giving charity with this expenditure, even if Hajj was obligatory upon her and her expenses were not incumbent upon you.