Hi, I have a nanny that I employ full time to take care of the house and my children while my wife and I are at work. Must we pay the nanny tax? Thank you

In this matter one must comply with the state laws and customs. (Shulchan Aruch Choshen Mishpat 331. See at length Sefer Mishpat Hapolim of Rabbi Yosef Rosner ZT"L, for the law and relationship between an employee and an employer according to Jewish law and the custom in our times).