In this work, we propose some efficient eigensolvers for computing the spectrum of the transmission eigenvalue problems which are derived from the Maxwell's equations with Tellegen media and pseudo-chiral media. For these problems, the quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed. Furthermore, we study the Helmholtz transmission eigenvalue problem for inhomogeneous anisotropic media with the index of refraction $n(x)\equiv 1$. For this problem, a mixed element scheme is presented, followed up with a sparse generalized eigenvalue problem whose size is so demanding even with a coarse mesh. We partially overcome this critical issue by deflating the almost all of the $\infty$ eigenvalue of huge multiplicity, resulting in a drastic reduction of the matrix size without deteriorating the sparsity.