Summary: In this paper, we apply Morse theory and local linking to study the existence of nontrivial solutions for Kirchhoff type equations involving the nonlocal fractional \(p\)-Laplacian with homogeneous Dirichlet boundary conditions: \[ \begin{cases} \left[ M\left( \iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right) \right]^{p-1}(-\Delta)_p^su(x)=f(x,u)\quad & \text{in}\,\, \Omega\text{,}\\u = 0\quad & \text{in}\,\, \mathbb{R}^N\setminus \Omega\text{,}\end{cases} \] where \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^N\), \((-\Delta)_p^s\) is the fractional \(p\)-Laplace operator with \(0<s<1<p<\infty\) with \(sp<N\), \(M \colon \mathbb{R}^{+}_{0}\rightarrow \mathbb{R}^{+}\) is a continuous and positive function not necessarily satisfying the increasing condition and \(f\) is a Carathéodory function satisfying some extra assumptions.