Fiber Bragg gratings (FBGs) have been widely employed as a sensor for temperature, vibration, strain, etc. measurements. However, extant methods for FBG interrogation still face challenges in the aspects of sensitivity, measurement speed, and cost. In this Letter, we introduced random speckles as the FBG's reflection spectrum information carrier for demodulation. Instead of the commonly used InGaAs cameras, a quadrant detector (QD) was first utilized to record the speckle patterns in the experiments. Although the speckle images were severely compressed into four channel signals by the QD, the spectral features of the FBGs can still be precisely extracted with the assistance of a deep convolution neural network (CNN). The temperature and vibration experiments were demonstrated with a resolution of 1.2 pm. These results show that the new, to the best of our knowledge, speckle-based demodulation scheme can satisfy the requirements of both high-resolution and high-speed measurements, which should pave a new way for the optical fiber sensors.