This study attempts to describe associated fluid dynamics of a square finite wall-mounted cylinder (FWMC) immersed within free-stream turbulent flow characterized by different turbulence intensities and integral length scales. An improved delayed detached eddy simulation method is adopted to numerically reproduce the fully developed turbulent flow fields. The results reveal that both the turbulence intensities and integral length scales have a significant effect on the separated shear layers, base pressure, and associated aerodynamic forces of the cylinder. Constrained streamlines along with critical point techniques are employed to further illustrate the influence of parameters of interest on a time-averaged flow pattern, including horseshoe vortex, surface flow, and wake topology. Distribution of second-order statistics within the wake region shows a shorter longitudinal length of the reversed flow region and enhanced vortex strength when background turbulence intensity increases. The time-dependent interaction between background turbulence and separated flow around the square FWMC is illustrated based on the phase difference between pressure of opposing side faces and the evolution of the reverse-flow region. In the end, the spectral proper orthogonal decomposition technique is employed to further investigate the effects of incoming flow turbulence on characteristics of the free-end shear flow and Von Kármán vortex shedding in the wake.