What is the ruling on a girl who swore falsely, with her hand on the Quran, saying: "By the life of this Quran, I did not do such and such...", and what is the expiation for that?

Swearing by the Mus'haf (copy of the Qur'an) constitutes a binding oath (yamin mun'aqidah) according to the majority of scholars, because the Qur'an is the Speech of Allah and an attribute of His attributes. Abu Hanifa and Ibn Hazm disagreed with this.

Swearing by the Mus'haf falsely is a false oath (yamin ghamus), which is a major sin for which there is no expiation, and what is obligatory in it is repentance.

It is permissible to lie in order to achieve a legitimate purpose that cannot be attained except through lying. So, if you swore falsely to maintain your relationship with your husband's family and there was no means except lying, then there is no sin upon you.