  This paper is devoted to revealing the relationship between matrix-valued
$\theta$-deformed bi-orthogonal polynomials and non-commutative Toda-type
hierarchies. In this procedure, Wronski quasi-determinants are widely used and
play the role of non-commutative $\tau$-functions. At the same time, B\"acklund
transformations are realized by using a moment modification method and
non-commutative $\theta$-deformed Volterra hierarchies are obtained, which
contain the known examples of the Itoh-Narita-Bogoyavlensky lattices and the
fractional Volterra hierarchy.
