A green and rapid method was developed to synthesize a series of carbon quantum dots (Fe,N/CQDs, Fe,S/CQDs, S,N/CQDs and Fe,S,N/CQDs) through thermal decomposition using persimmon frost as a carbon source. All the synthesized CQDs have peroxidase-like activity, and the comparision of their steady-state kinetic parameters inferred that the peroxidase-like activity of Fe,S,N/CQDs is superior to those of Fe,S/CQDs and Fe,N/CQDs. Furthermore, Fe,S,N/CQDs exhibited good affinity towards 3,3′,5,5′-Tetramethylbenzidine (TMB) and H2O2 compared to natural horseradish peroxidase (HRP) and other nanozymes. In theory, uric acid (UA) can produce H2O2 with the catalysis by uricase. Fe,S,N/CQDs could catalyze the breakdown of H2O2 to produce reactive oxygen species (ROS), which can oxidize TMB (colorless) to oxTMB (blue). Based on the nanozyme-catalyzed oxidation of TMB, a simple and sensitive colorimetric sensor was developed for the quantitative analysis of H2O2 and UA. Low limit of detection (0.73 μM), wide linear range (1–80 μM) and good repeatability were obtained. The sensor was also demonstrated for the determination of UA in biological samples and satisfactory results were obtained.