We examine both optical and optical/near-infrared (NIR) color distributions of the globular cluster (GC) system in the core of the Coma cluster of galaxies (Abell 1656), centered on the giant elliptical galaxy NGC 4874, to study how non-linearities in the color-metallicity relations of GC systems in large elliptical galaxies are linked to bimodal optical color distributions. Since optical-NIR color distributions of extragalactic GC systems reflect the underlying features of the metallicity distributions, we also present the color-color relation for this GC system. In order to do this, we combine F160W ($H_{160}$) NIR imaging data acquired with the Wide Field Camera 3 IR Channel (WFC3/IR), newly installed on Hubble Space Telescope (HST), with F475W ($g_{475}$) and FF814W ($I_{814}$) optical imaging data from the HST Advanced Camera for Surveys (ACS). To quantitatively explain the feature of color distributions, we use the Gaussian Mixture Modeling (GMM) code. Finally, we show the radial distribution of the GCs in the field of NGC 4874.