How can new Software QA processes be introduced in an existing organization?

A lot depends on the size of the organization and the risks involved. For large organizations with high-risk (in terms of lives or property) projects, serious management buy-in is required and a formalized QA process is necessary.

Where the risk is lower, management and organizational buy-in and QA implementation may be a slower, step-at-a-time process. QA processes should be balanced with productivity so as to keep bureaucracy from getting out of hand.
For small groups or projects, a more ad-hoc process may be appropriate, depending on the type of customers and projects. A lot will depend on team leads or managers, feedback to developers, and ensuring adequate communications among customers, managers, developers, and testers.
 

The most value for effort will often be in
(a) requirements management processes, with a goal of clear, complete, testable requirement specifications embodied in requirements or design documentation, or in 'agile'-type environments extensive continuous coordination with end-users, 

(b) design inspections and code inspections, and 

(c) post-mortems/retrospectives.