That sentence is not, in fact, a full sentence. It is called a "fragment", often used in literature for dramatic effect. The full sentence would read:
"She presses the subject of the memoir, if only to put to rest the perforated eardrum question."
The "perforated eardrum question" will be, indeed, an unfinished story about somebody's eardrum being perforated. As Randy said, if such a story is "put to rest", then it is concluded, no more questions are left to be asked.