Fixed-rate quantizers whose bin levels are adaptive have been used in the networked control literature as efficient schemes for stabilizing open-loop unstable noise-free linear systems with arbitrary initial conditions connected over noiseless channels. In this note, stochastic stability results for such simple adaptive quantizers when the system noise has unbounded support for its probability measure are presented. It is shown that, there exists a unique invariant distribution for the state and the quantizer parameters under mild conditions. The second moment under the invariant distribution is finite, if the system noise is Gaussian.