Event-driven spiking neural networks(SNNs) are promising neural networks that reduce the energy consumption of continuously growing AI models. Recently, keeping pace with the development of transformers, transformer-based SNNs were presented. Due to the incompatibility of self-attention with spikes, however, existing transformer-based SNNs limit themselves by either restructuring self-attention architecture or conforming to non-spike computations. In this work, we propose a novel transformer-to-SNN conversion method that outputs an end-to-end spike-based transformer, named SpikedAttention. Our method directly converts the well-trained transformer without modifying its attention architecture. For the vision task, the proposed method converts Swin Transformer into an SNN without post-training or conversion-aware training, achieving state-of-the-art SNN accuracy on ImageNet dataset, i.e., 80.0\% with 28.7M parameters. Considering weight accumulation, neuron potential update, and on-chip data movement, SpikedAttention reduces energy consumption by 42\% compared to the baseline ANN, i.e., Swin-T. Furthermore, for the first time, we demonstrate that SpikedAttention successfully converts a BERT model to an SNN with only 0.3\% accuracy loss on average consuming 58\% less energy on GLUE benchmark.