Actually, your first answer should be "progress".
Progress means (in simple terms) to "move forward". Any medical condition usually progresses - that means it usually gets worse or better.
Convert means to change from one thing to another. A cough is not going to suddenly "change" to shortness of breath. It will take some time.
So your sentence should read:
Within a week, the patient had a dry cough, which might progress to shortness of breath. In 10% to 20% of cases, patients require automatic ventilation to breathe.
Hope that helps!