We develop a fiber Bragg grating accelerometer based on a bearing and flexure hinge for the measurement of medium-high frequency vibration signals. The mathematical model between its natural frequency and sensitivity and structural parameters is derived based on a mechanical model, and the structural design is optimized based on the theoretical analysis results. With these prerequisites, the sensor was fabricated. Ultimately, its dynamic characteristics are validated using a finite element simulation and vibration experiment. The results show that both its operating frequency range and acceleration sensitivity are 10-1200 Hz and 17.25 pm/g. In addition, this proposed sensor has some advantages such as an error of less than 0.3 g, a good linearity of greater than 0.99, a repeatability error of 2.33%, and it is free of temperature.