In my experience typically food quality and safety complaints are ultimately handled by the QA and/or QC Manager, so complaints received by Customer Service or anyone else need to forward the complaint to the Quality Manager and this should be written as a SOP.
Next, an investigation including root cause analysis conducted in order to determine possible causes of the issue including such tools as interviews with pertinent production, shipping, sanitation, maintenance employees, etc. along with record reviews, root cause analysis tools if possible. If a root cause analysis points to a specific issue and/or set of issues) that may have occurred and likely caused the issue resulting in a complaint, corrective and/or preventive actions are implemented. This info should then be included in a concise report and sent to your customer, and everything should ideally be documented on a spreadsheet for quick access to show complaint trend analyses, reviews at food quality and safety meetings, your annual management reviews, etc.