We experimentally conduct Brillouin dynamic grating (BDG) operation using a 1-km-long four-mode fiber. By employing a simplified ring-cavity configuration with single-end pumping, the BDG is effectively generated in $ {{\rm LP}_{01}} $LP01 mode within a range of 250 m, and three higher-order modes, namely, $ {{\rm LP}_{11b}} $LP11b, $ {{\rm LP}_{21a}} $LP21a, and $ {{\rm LP}_{02}} $LP02, are chosen as probes to analyze the BDG with a spatial resolution of 1 m. To the best of our knowledge, this is the first time to characterize the responses of BDG frequency to temperature and strain for different modes in a conventional few-mode fiber. By employing the pump-probe pair of $ {{\rm LP}_{01}}{{\rm - LP}_{02}} $LP01-LP02 mode, the highest temperature and strain sensitivities of 3.21 MHz/°C and $ - 0.0384\;{\rm MHz}/{\unicode{x00B5}}{\unicode{x03B5}} $-0.0384MHz/µε have been achieved. Also, the performance of simultaneously distributed temperature and strain sensing based on BDG is evaluated.