In fiber Bragg grating (FBG) sensor networks, the highly overlapped spectral signals can lead to considerable errors in center wavelength demodulation. To tackle this problem, we utilize the fully convolutional time-domain audio separation network (Conv-TasNet) model to produce a distinct spectral signal, which is then demodulated using the dual-weight centroid approach to determine the spectral signal’s center wavelength. Specifically, we first demonstrate the theoretical feasibility of the Conv-TasNet model on simulated data. Experimental results show that the Conv-TasNet model can separate the signals of three FBG sensors. After that, we collect the spectral data and further train and validate the model based on the pretrained model of the simulated data to see how it performs on the real data. The experiments consistently illustrate superior performance of our Conv-TasNet model that can also separate actual spectrum signals. The same performance can be achieved by applying the pretrained model but with less training data. The model obtains a competitive performance compared to currently available methods. Moreover, the method provides a solution for improving the multiplexing performance of the FBG sensor network.