In this paper, we consider rate-distortion tradeoff problems for time-varying, multi-dimensional, partially observable Gauss-Markov processes subject to the zero-delay constraint. As a distortion metric, we consider the mean square error between the hidden state process and the reconstructed process. It is shown that an optimal test channel can be realized by a cascade connection of a pre-Kalman filter estimating the hidden state of the Gauss-Markov process, an additive white Gaussian noise channel, and a post-Kalman filter estimating the internal state of the pre-Kalman filter. An optimal test channel can be constructed by semidefinite programming (SDP). We also show that for stationary sources, there exists a time-invariant optimal test channel, which can also be found by SDP.