This paper is concerned with the characterization of quadratic mean square (QMS) stabilizing controllers for systems with both quantization error and multiplicative random noise. An observer-based output feedback controller is proposed and designed to achieve the QMS stability of the closed-loop system, and all QMS stabilizing controllers in this structure are characterized in terms of an inequality involving the Mahler measure of the plant, together with an algebraic Riccati inequality. It is noted that the present results potentially pave the road to search for solutions to a more specific control design to address additional system performance requirements.