Using newly developed $\mathbf{H}$(${\rm curl}^2$) conforming elements, we solve the Maxwell’s transmission eigenvalue problem. Both real and complex eigenvalues are considered. Based on the fixed-point weak formulation with reasonable assumptions, the optimal error estimates for numerical eigenvalues and eigenfunctions (in the $\mathbf{H}$(${\rm curl}^2$))-norm and $\mathbf{H}$(${\rm curl}$)-semi-norm) are established. Numerical experiments are performed to verify the theoretical assumptions and confirm our theoretical analysis.