Let $\Omega$ be a bounded domain in $\mathbb{R}^N, N\geq 2,$ with smooth boundary $\Sigma$ and let $\Omega_1$ be a subdomain of $\Omega$ with smooth boundary $\Gamma,$ such that $\overline{\Omega}_1\subset \Omega$. Denote $\Omega_2 = \Omega \setminus \overline{\Omega}_1.$ Consider the transmission eigenvalue problem \begin{equation*} \left\{\begin{array}{l} -\Delta_p u_1+\gamma_1(x)\mid u_1\mid ^{r-2}u_1=\lambda \mid u_1\mid ^{p-2}u_1\ \ \mbox{in} ~ \Omega_1,\\[1mm] -\Delta_q u_2+\gamma_2(x)\mid u_2\mid ^{s-2}u_2=\lambda \mid u_2\mid ^{q-2}u_2\ \ \mbox{in} ~ \Omega_2,\\[1mm] u_1=u_2,~~\frac{\partial u_1}{\partial\nu_{p}}=\frac{\partial u_2}{\partial\nu_{q}} ~~ \mbox{on} ~ \Gamma,\\[1mm] \frac{\partial u_2}{\partial\nu_{q}}+\beta (x) \mid u_2\mid^{\zeta-2} u_2=0 ~~ \mbox{on} ~ \Sigma, \end{array}\right. \end{equation*} where $\lambda$ is a real parameter $p, q, r, s, \zeta \in (1, \infty)$ and $\gamma_i\in L^{\infty}(\Omega_i), ~i=1, 2, \beta\in L^{\infty}(\Sigma),$ $\beta\geq 0$ a.e. on $\Sigma.$ Under additional suitable assumptions on $p, q, r, s, \zeta$ we prove the existence of a sequence of eigenvalues $\big(\lambda_n\big)_n, \lambda_n\rightarrow \infty.$ The proof is based on the Lusternik-Schnirelmann theory on $C^1-$ manifolds.