Is it permissible to speak about the bad actions of a female relative of mine without mentioning her name, and is that considered backbiting?

Your mentioning to one of your female cousins what she dislikes falls under the category of backbiting, which the Prophet, peace and blessings be upon him, defined as: "Your mentioning about your brother what he dislikes." And your mentioning that she is "one of your female cousins" is a specification close to identification, and it may lead to ill-suspicion about all your female cousins, as long as the listeners do not know which one of them is meant by specific identification.