Is the statement: "Does Allah lie?" considered ill-mannered towards Allah and does its utterer incur sin?

Piety towards Allah means magnifying His sanctities and rituals. Your question to your mother about the Book of Allah was an expression of disapproval, which is good. However, the discussion with your mother should have been in a gentle manner, because asking a question in a harsh way can be perceived as rudeness and severity, which is not appropriate. Your mother's dislike of the Quran's ruling is impermissible, but her statement can be interpreted in the best possible light, meaning she did not intend to object to Allah's decree.