What regulations govern the quality management system for producers of aircraft components?
From FAA, Production Certificate:


  A production certificate is an approval (document) to manufacture
  duplicate products under an FAA-approved type design (e.g., type
  certificate or supplemental type certificate). The holder of a
  production certificate may obtain an airworthiness certificate for
  aircraft produced under the production certificate without further
  showing. To obtain one, the production application and approval
  process must be followed.


See also: 
Production Approval - Regulations &amp; Policies (link).