The classical Weyl quantization formula \[ (W(f)u)(x)=\int_{R^ n}\int_{R^ n}f\left ({x+y\over 2},\zeta\right) e^{2\pi i(x-{y\over \zeta})}u(y)dyd\zeta \] is extended to the form \[ W(f)=\int_ \Omega f(\zeta)W(s_ \zeta)d\mu_ \Omega(\zeta) \] where \(\Omega\) is a bounded symmetric domain with invariant measure \(\mu_ \Omega\) (the author specializes here on the hyperbolic matrix ball \[ \Omega=\{z\in\mathbb{C}^{r\times r}: I-zz^*>0\}, r>0) \] and \(W\) is a unitary representation of the group of symmetries of \(\Omega\). The main result (Theorem 5.5) provides conditions on the symbol function \(f\) under which \(W(f)\) is a bounded Hilbert space operator.