Ultraviolet (UV) quantum emitters in hexagonal boron nitride (hBN) have generated considerable interest due to their outstanding optical response. Recent experiments have identified a carbon impurity as a possible source of UV single-photon emission. Here, on the basis of first-principles calculations, we systematically evaluate the ability of substitutional carbon defects to develop the UV color centers in hBN. Of 17 defect configurations under consideration, we particularly emphasize the carbon ring defect (6C), for which the calculated zero-phonon line agrees well the experimental 4.1 eV emission signal. We also compare the optical properties of 6C with those of other relevant defects, thereby outlining the key differences in the emission mechanism. Our findings provide new insights into the strong response of this color center to external perturbations and pave the way to a robust identification of the particular carbon substitutional defects by spectroscopic methods.