I know for sure that BMI(Body Mass Index) is a quantitative variable as it is a continuous variable. But is that BMI Category derived from the BMI a qualitative variable or a quantitative variable? (Underweight, Normal Weight, Overweight. Thanks
Usually we think of this type of data as a special form of categorical data called "ordinal", that is, ordered-categorical. This is because there is a natural ordering in the data: Underweight < Normal < Overweight.
While it is sometimes useful to create categories such as these, there is a great loss of information by doing so.
Edit: Based on comments in the question, it would be a good idea to consider a regression based model, where you do not categorise BMI but rather use it as the outcome/response. Since you seem to have a just one independent variable, which is categorical, this would be a one-way ANOVA.