Sequential rate-distortion (SRD) theory provides a framework for studying the fundamental trade-off between data-rate and data-quality in real-time communication systems. In this paper, we consider the SRD problem for multi-dimensional time-varying Gauss-Markov processes under mean-square distortion criteria. We first revisit the sensor-estimator separation principle, which asserts that considered SRD problem is equivalent to a joint sensor and estimator design problem in which data-rate of the sensor output is minimized while the estimator's performance satisfies the distortion criteria. We then show that the optimal joint design can be performed by semidefinite programming. A semidefinite representation of the corresponding SRD function is obtained. Implications of the obtained result in the context of zero-delay source coding theory and applications to networked control theory are also discussed.