No, a graduate tax will not increase the quality of schools, because taxing the earnings of students takes the value out of going to college. The students will have the same standard of living they would have before college, because they will be paying all of their extra money in taxes, rather than enjoying the benefits of their hard work.
It is already hard enough to go to school. Many adults want to go to school, but they can't find the time because they are working and raising a family. Many school graduates are burdened with thousands of dollars of student loans that are difficult to repay. A graduate tax would only make it harder to go to school and reduce school enrollment.