Quality planning is the responsibility of the project manager, and the level of responsibility varies by company and company culture after that. Often there is a functional QA manager that has the overall responsibility for quality.
The response to quality issues will vary based on the significance of the non-conformance. If minor, just make a correction and go on. If major, probably best to inform the key stakeholders along with your plan of corrective action. Make your handling of recovery impressive, decisive, and clearly communicated to leave the best impression.