Summary: We establish an existence and uniqueness results for a homogeneous Neumann boundary value problem involving the \(p(x)\)-Kirchhoff-Laplace operator of the following form \[ \begin{cases} \begin{aligned} &-M\Big (\int_{{\Omega}}\frac{1}{p(x)}(\vert \nabla u\vert^{p(x)}+\vert u\vert^{p(x)})\,dx\Big ) \Big (\text{{div}}(\vert \nabla u\vert^{p(x)-2}\nabla u)-\vert u\vert^{p(x)-2}u\Big )\\ &=f(x, u, \nabla u) && \text{in } , \\ &\vert \nabla u\vert^{p(x)-2}\frac{\partial u}{\partial \eta }=0 && \text{on } \partial{{\Omega}}. \end{aligned} \end{cases} \] where \({{\Omega}}\) is a smooth bounded domain in \(\mathbb{R}^N\), \(\frac{\partial u}{\partial \eta }\) is the exterior normal derivative, \(p(x)\in C_+({\overline{{\Omega} }})\) with \(p(x)\ge 2\). By means of a topological degree of Berkovits for a class of demicontinuous operators of generalized \((S_+)\) type and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on \(f\) and \(M\), we obtain a results on the existence and uniqueness of weak solution to the considered problem.