For me it depends on the role/responsibilities of the quality manager as determined by your organizational structure.
If you're specifically talking about the adherence to pre-defined quality standards (project quality management), that is a project specific measurement.
Therefore, it makes sense to me that the quality manager would manage/monitor project quality, but would ultimately work under the direction of the PM.
The project manager being the “face” for the project, and ultimately responsible for the success or failure of the project as a whole.
With that being said, everyone on the team should understand that they are responsible for quality.
On the other hand, if you're talking about the entire quality process (definition, planning, management/measurement), then that could be "owned" by multiple individuals or groups both within and external (i.e. the client) to your organization.
Some elements of which could be the sole responsibility of the quality manager and his or her department. Again, depending on the size and structure of your organization. Saving Changes...