The two terms really have nothing to do with one another. Liberal education is a term, taken form the term liberal arts education. Which means a college education that is founded in a variety of subjects to give the student a well rounded education. They get a taste of many subjects, such as science, philosophy, biology, art, communication, history ect.
A quality education is certainly different and must be considered an opinion. Many people feel that a liberal arts education is a quality education as it gives you wide knowledge on many subjects, educated. You may then decide to specialize in your Masters work.
Hope that helps.