The analysis of hydrophobicity holds great importance in understanding the water-repelling characteristics exhibited by molecules, surfaces, and materials. This research delves into the influence of the strength of solid-fluid interactions on the wetting and hydrophobic nature of rough surfaces. Employing a two-dimensional (2D) pseudo-potential multiphase lattice Boltzmann method with a D2Q9 model, the investigation focuses on the surface hydrophobicity using an equilibrium state of a water droplet on a flat. The analysis of hydrophobicity on rough surfaces involves the use of droplets with a radius of 150 lattice units within a 1000x1000 lattice unit domain. The findings indicate a significant augmentation in hydrophobicity as the solid-fluid interaction parameter is heightened, leading to an increased contact area between water droplets and solid walls. The assessment of hydrophobicity relies on measuring the contact angle at the interface between the solid and fluid-vapor. Furthermore, the study underscores the influence of average roughness and saturation temperature on surface hydrophobicity.