There are many possible questions that could be explored in many different areas of the social sciences. Let us look at a possible research question from political science.
Our research question could be "has the American political system become more polarized in the past 40 years?" This question could include some quantitative research, but it would also include a great deal of qualitative study.
As part of the qualitative work, a researcher might look at State of the Union addresses and try to determine if their tone has changed over the time period in question. The researcher could also interview major political players and observers of the political process to try to determine their feelings on the subject. These would both be qualitative methods because they would involve in-depth, verbal data that would not lend itself well to being put into numerical form.
There are, of course, practically an infinite number of other issues that could be researched by qualitative methods.