We propose and develop a high-sensitivity fiber Bragg grating (FBG) accelerometer based on a flexible hinge for vibration sensing. The proposed method employs a 1-mm FBG that is inscribed pointwise using a femtosecond laser such that the resultant stress concentration at the FBG is improved significantly by reducing the distance between the two bonding points of the FBG structure. Through theoretical calculations and finite element simulations, the relationship between sensitivity and distance was found to be inversely proportional. Using this structure, the sensitivity was enhanced to 804 pm/g at 15 and 30 Hz. The resonant frequency of the proposed accelerometer is 102 Hz, and the working frequency bandwidth is from 10 to 50 Hz. The performance of the accelerometer allows the detection of vibrations in seismic explorations with high reliability.