The complex consimilarity of complex matrices defined by \(\overline{P}^{-1} AP= B\), where \(A,B,P\in \mathbb{C}^{n\times n}\) and \(P\) is invertible, is not extensible to the quaternions since \(\overline{AB}\neq \overline{AB}\) in general. Thus, given quaternion matrices \(A,B\in \mathbb{H}^{n\times n}\), the author defines the \(j\)-conjugate of \(A\) and consimilarity of matrices \(A\) and \(B\), respectively, in the following way: \(\widetilde{A}=-{\mathbf j}A{\mathbf j}\) and \(\widetilde{P}^{-1}AP= B\). This is a natural extension of complex consimilarity of complex matrices, since if \(P\in\mathbb{C}^{n\times n}\), then \(\widetilde{P}= \overline{P}\). The author defines also the right coneigenvalue problem \(A\widetilde{X}= X\lambda\) where \(A\in \mathbb{H}^{n\times n}\), \(0\neq X\in \mathbb{H}^n\) and \(\lambda\in \mathbb{H}\). As a result of this definition he obtains a series of results similar to the usual right eigenvalue problem of quaternion matrices and establishes the relation between right coneigenvalues and right eigenvalues of quaternion matrices. This definition of consimilarity of quaternion matrices has many good properties including some that are essentially different from those of complex consimilarity.