What might be some appropriate research questions within a qualitative study for doctoral research?
Within the broad category of social sciences research there are millions of research questions that could be asked in a qualitative study for your dissertation. The specifics depend so much on what your actual dissertation is about.
In political science, you might ask whether African Americans (or other minorities) are actually helped by having members of their race in positions of power in city government. This would be qualitative in many ways because there are many ways to define "benefits" that might come from being represented.
In sociology, you might ask about whether people with more education have a higher quality of life. Here too, there would be qualitative aspects to "quality of life" that could not be measured simply by using numbers.
You might also ask whether standardized testing improves the quality of education that students receive. This would be qualitative because there would be issues pertaining to what constitutes a quality education that could not be measured quantitatively.
These are the sorts of research questions that could be part of a qualitative research project in the social sciences.