For a vector Markov source driven by additive Gaussian noise, we study the existence and structure of optimal quantization policies. The quantizers allowed are the ones which have convex codecells. For the finite horizon problem and bounded cost, we show that an optimal zero-delay quantization policy exists. Then the linear quadratic Gaussian problem is considered as an important extension of the bounded cost assumption for the finite horizon setting. For the infinite horizon setup, the existence of an optimal stationary policy is established among the class of Markov coding policies.