For any $A=A_1+A_2j\in \bold Q^{n\times n}$ and $\eta\in\{i,j,k\}$, denote $A^{\eta H}=-\eta A^H\eta$. If $A^{\eta H}=A$, $A$ is called an $\eta$-Hermitian matrix. If $A^{\eta H}=-A$, $A$ is called an $\eta$-anti-Hermitian matrix. Denote $\eta$-Hermitian matrices and $\eta$-anti-Hermitian matrices by $\eta\bold{HQ}^{n\times n}$ and $\eta\bold{AQ}^{n\times n}$, respectively. In this paper, we consider the least squares $\eta$-Hermitian problems of quaternion matrix equation $A^HXA+B^HYB=C$ by using the complex representation of quaternion matrices, the Moore--Penrose generalized inverse and the Kronecker product of matrices. We derive the expressions of the least squares solution with the least norm of quaternion matrix equation $A^HXA+B^HYB=C$ over $[X,Y]\in\eta\bold{HQ}^{n\times n}\times\eta\bold{HQ}^{k\times k}$, $[X,Y]\in\eta\bold{AQ}^{n\times n}\times\eta\bold{AQ}^{k\times k}$ and $[X,Y]\in\eta\bold{HQ}^{n\times n}\times\eta\bold{AQ}^{k\times k}$, respectively.