  A $C^*$-textile dynamical system $({\cal A},
\rho,\eta,\Sigma^\rho,\Sigma^\eta, \kappa)$ connsists of a unital $C^*$-algebra
${\cal A}$, two families of endomorphisms ${\rho_\alpha}_{\alpha \in
\Sigma^\rho}$ and ${\eta_a}_{a \in \Sigma^\eta}$ of ${\cal A}$ and certain
commutation relations $\kappa$ among them. It yields a two-dimensional subshift
and multi structure of Hilbert $C^*$-bimodules, which we call a Hilbert
$C^*$-quad module. We introduce a $C^*$-algebra from the Hilbert $C^*$-quad
module as a two-dimensional analogue of Pimsner's construction of
$C^*$-algebras from Hilbert $C^*$-bimodules. We study the $C^*$-algebras
defined by the Hilbert $C^*$-quad modules and prove that they have universal
properties subject to certain operator relations. We also present its examples
arising from commuting matrices.
