Under noiseless channel conditions and for sources with memory, finite-state vector quantizers (FSVQs) exhibit improvements over memoryless vector quantizers. It is shown, however, that in the presence of channel noise, the performance of the FSVQ degrades significantly. This suggests that for noisy channels, the FSVQ design problem needs to be reformulated by taking into account the channel noise. Using some of the developments in joint source-channel trellis coding, we describe two different methods leading to two types of noisy channel FSVQs. We show by means of simulations on the Gauss-Markov source and speech LSP parameters and for a binary symmetric channel that both schemes are fairly robust to channel noise. For the Gauss-Markov source, the proposed noisy channel FSVQs perform at least as well as or better than the channel-optimized VQ, while for speech LSP parameters, they lead to saving of 1.5-4 bits/frame over the channel-optimized VQ depending on the level of noise in the channel.