What is the difference between defending an argument and qualifying an argument?
Both "defend" and "qualify" have several definitions. In the uses you are questioning, defending an argument refers to supporting a position. When a person defends his or her argument, s/he is attempting to support his/her position; you defend an argument by presenting evidence that helps to prove your stance.
Qualifying an argument refers to providing more information. One qualifies an argument by explaining, in more or greater detail, the qualities or characteristics that are relevant to the topic.
So, defending an argument is supporting a specific side of the discussion. Qualifying an argument involves adding information about the subject but doesn't necessarily include taking a position for or against the topic.