Summary: We study the multiplicity of non-negative solutions for the quasilinear \(p\)-Laplacian equation with the nonlinear boundary condition \[ \begin{cases}-\Delta_pu+a(x)|u|^{p-2}u=\lambda h(x)|u|^{q-2}u,\quad & x \in\Omega,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial v}=g(x)|u|^{r-2}u,\quad & x\in\partial\Omega,\end{cases}\tag{1} \] where \(\Delta_p\) denotes the \(p\)-Laplacian operator, defined by \(\Delta_pu=\text{div} (|\nabla u|^{p-2}\nabla u)\), \(1<p<N\), \(\Omega\) is a smooth exterior domain in \(\mathbb{R}^N(N>1)\). \(\frac{\partial}{\partial v}\) is the outward normal derivative, \(\lambda\in\mathbb{R}^1\setminus\{0\}\). The parameters \(p,q,r\) are either \(1<r<p<q<p^*=\frac{Np}{N-p}\) or \(1<q<p<r<p_0= \frac{p(N-1)}{N-p}\). The weight functions \(a(x),h(x),g(x)\) satisfy some suitable conditions. Using the decomposition of the Nehari manifold and the variational methods, we prove that problem (1) has at least two positive solutions provided \(0<|\lambda|<\lambda_1\) for some \(\lambda_1\).