Transformer is an emerging neural network model with attention mechanism. It has been adopted to various tasks and achieved a favorable accuracy compared to CNNs and RNNs. While the attention mechanism is recognized as a general-purpose component, many of the Transformer models require a significant number of parameters compared to the CNN-based ones. To mitigate the computational complexity, recently, a hybrid approach has been proposed, which uses ResNet as a backbone architecture and replaces a part of its convolution layers with an MHSA (Multi-Head Self-Attention) mechanism. In this paper, we significantly reduce the parameter size of such models by using Neural ODE (Ordinary Differential Equation) as a backbone architecture instead of ResNet. The proposed hybrid model reduces the parameter size by 94.6% compared to the CNN-based ones without degrading the accuracy. We then deploy the proposed model on a modest-sized FPGA device for edge computing. To further reduce FPGA resource utilization, we quantize the model following QAT (Quantization Aware Training) scheme instead of PTQ (Post Training Quantization) to suppress the accuracy loss. As a result, an extremely lightweight Transformer-based model can be implemented on resource-limited FPGAs. The weights of the feature extraction network are stored on-chip to minimize the memory transfer overhead, allowing faster inference. By eliminating the overhead of memory transfers, inference can be executed seamlessly, leading to accelerated inference. The proposed FPGA implementation achieves 12.8x speedup and 9.21x energy efficiency compared to ARM Cortex-A53 CPU.