Non-invasive brain-computer interfaces help the subjects to control external devices by brain intentions. The multi-class classification of upper limb movements can provide external devices with more control commands. The onsets of the upper limb movements are located by the external limb trajectory to eliminate the delay and bias among the trials. However, the trajectories are not recorded due to the limitation of experiments. The delay cannot be avoided in the analysis of signals. The delay negatively influences the classification performance, which limits the further application of upper limb movements in the brain-computer interface. This work focuses on multi-channel brain signals analysis in the temporal-frequency approach. It proposes the two-stage-training temporal-spectral neural network (TTSNet) to decode patterns from brain signals. The TTSNet first divides the signals into various filter banks. In each filter bank, task-related component analysis is used to reduce the dimension and reject the noise of the brain. A convolutional neural network (CNN) is then used to optimize the temporal characteristic of signals and extract class-related features. Finally, these class-related features from all filter banks are fused by concatenation and classified by the fully connected layer of the CNN. The proposed method is evaluated in two public datasets. The results show that TTSNet has an improved accuracy of 0.7456$\pm$0.1205 compared to the EEGNet of 0.6506$\pm$0.1275 ($p<0.05$) and FBTRCA of 0.6787$\pm$0.1260 ($p<0.1$) in the movement detection task, which classifies the movement state and the resting state. The proposed method is expected to help detect limb movements and assist in the rehabilitation of stroke patients.