Many companies ask the Quality Officer to write quality documentation.
In my opinion, smart companies ask the Quality Officer, or a Lead Implementer to work together with users to write quality documentation. For example, work with people for Purchasing and Warehouse to write procedures regarding ordering and receiving supplies.
Asking someone to write the quality documentation only because it is an internal auditor is not common, and I think, not useful.
You can find more information in the following links:
First Identify the Process Owners - get the process owners to document their processes. the Quality dept should be able to map these into procedures and documents as required by the standard - dont forget the top management of course