Studies on integrating Spiking Neural Networks (SNNs) with the Transformer architecture holds promise for enabling models to achieve ultra-low energy consumption while possessing the performance of the Transformer architecture. Currently, studies on ANN-to-SNN conversion of integrating Spiking Neural Networks (SNNs) with the Transformer architecture mainly focuses on simple activation functions in MLPs, and has not yet addressed the mismatch between the Softmax activation function in the self-attention mechanism and the computation rules of SNNs. Consequently, the ANN-to-SNN conversion efforts have consistently failed to make the Transformer architecture directly applicable to SNNs. To address this challenge, we propose the Spiking-Softmax method, which integrates Spiking Exponential Neuron (SI-exp) and Spiking Collaboration Normalized Neuron (SI-norm). The Spiking-Softmax method accurately simulates the Softmax activation function with only 12 time steps. Building upon this, we propose the Spike Integrated Transformer conversion (SIT-conversion) method, which enables the conversion of the Transformer architecture to SNNs. The SNNs generated by the SIT-conversion of Transformer models of various sizes achieve accuracy nearly identical to their ANNs counterparts, achieving nearly lossless and ultra-low-latency ANN-to-SNN conversion. This work represents the first implementation of simulating the Softmax activation function and fully converting the Transformer architecture into SNNs through spike firing.