Is a shallow cough a symptom of ovarian cancer?

Very unlikely: A shallow cough has many more likely causes than ovarian cancer. The most common symptom of ovarian cancer is increased abdominal fullness, and unfortunately it is a late sign.

Nothing definite.: It could be if the cancer has spread to the chest, in which case it would likely have caused many, many symptoms in the abdomen like increased girth, heaviness, bloating, bowel or bladder or vaginal discomfort.