In this paper, we study the Integrated Communication and Control (ICAC) problem. Specifically, we investigate how messages can be transmitted from the controller/encoder to the observer/decoder through the control signal in Multiple-Input Multiple-Output (MIMO) vector-state Linear Quadratic Gaussian (LQG) systems under control constraints. We provide a computable capacity expression using semidefinite programming. We further show that it is possible to transmit data at a nonzero rate over an LQG system while maintaining the same optimal control cost as in the case where no information message are transmitted. Finally, we discuss how this framework generalizes communication over MIMO Gaussian channels with feedback, both with and without InterSymbol Interference (ISI).