Vector Quantization in Hidden Markov Models
do multiple HMM classifiers (one for each class) share common Codebook
  (i.e common set of symbols) in Vector Quantization?


Yes, for example in large vocabulary speech recognition there is a big codebook of about 4000-10000 gaussians which are shared across 20-30k HMMs


  To which Codebook the test input vector belongs to (we need this to
  determine the symbol, in turn which is fed to the HMM for
  classification)?


In modern ASR systems codebook defines a set of GMM probability distributions, not the discrete symbols. So "belongs" doesn't really have sense, it's more a probability value. But during the classification a test vector is usually evaluated for the subset of all codebooks which is active for this stage.

Active codebooks are tracked across test vectors based on active HMMs and preceeding history. Whole HMM set is pruned in order to get active ones. Usually there are about 10k active HMMs on each step and about 1k codebooks.