To solve the spectral signal overlapping problem in fiber Bragg grating (FBG) sensing networks, this study proposes a spectral signal demodulation algorithm using a convolutional neural network model based on a field-programmable gate array (FPGA) and implements it in hardware for acceleration. The models parameters are quantized to a fixed-point representation, reducing the storage space of the model and enhancing the utilization of DSP resources in the FPGA. Hardware optimization techniques such as loop unrolling and array rearrangement are employed to improve real-time system performance, establishing a parallel computing scheme for the algorithm. The results indicate that under a clock frequency of 100 MHz, the demodulation accuracy of the test set is 1.19pm at an inference speed of 14.96μs per frame and a spectral demodulation rate of 60kHz. The proposed algorithm exhibits high precision and speed in the demodulation of overlapped FBG spectral signals.