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

Matrix-valued $\theta$-deformed bi-orthogonal polynomials, Non-commutative Toda theory and B\"acklund transformation