A square matrix $A$ is called a Hurwitz matrix if all eigenvalues of $A$ have strictly negative real part, $Re[\lambda_i] < 0$ $A$ is also called a stability matrix, because the feedback system $$ \dot x = A x $$ is stable.

If $G(s)$ is a (matrix-valued) transfer function, then $G$ is called Hurwitz if the poles of all elements of $G$ have negative real part. Note that it is not necessary that $G(s)$ for a specific argument $s$ be a Hurwitz matrix -- it need not even be square. The connection is that if $A$ is a Hurwitz matrix, then the dynamical system \begin{eqnarray*} \dot x(t) &=& A x(t) + B u(t) \\ y(t) &=& C x(t) + D u(t) \end{eqnarray*}has a Hurwitz transfer function.

Reference: Hassan K. Khalil, Nonlinear Systems, Prentice Hall, 2002