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The purpose of this article is to establish the molecular decomposition of the homogeneous Besov-Morrey spaces associated with the Hermite operator $\mathbb{H} = -\Delta+|x|^2$ on the Euclidean space $\mathbb{R}^n$. Particularly, we obtain some estimates for the operator $\mathbb{H}$ on the Hermite-Besov-Morrey spaces and the regularity results to the fractional Hermite equations $$ (-\Delta +|x|^2 )^su=f, $$ and $$ (-\Delta +|x|^2 +I)^su=f. $$ Our results generalize some results by Anh and Thinh [1].