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Spiking neural networks (SNNs) have received increasing attention due to their high biological plausibility and energy efficiency. The binary spike-based information propagation enables efficient sparse computation in event-based and static computer vision applications. However, the weight precision and especially the membrane potential precision remain as high-precision values (e.g., 32 bits) in state-of-the-art SNN algorithms. Each neuron in an SNN stores the membrane potential over time and typically updates its value in every time step. Such frequent read/write operations of high-precision membrane potential incur storage and memory access overhead in SNNs, which undermines the SNNs' compatibility with resource-constrained hardware. To resolve this inefficiency, prior works have explored the time step reduction and low-precision representation of membrane potential at a limited scale and reported significant accuracy drops. Furthermore, while recent advances in on-device AI present pruning and quantization optimization with different architectures and datasets, simultaneous pruning with quantization is highly under-explored in SNNs. In this work, we present SpQuant-SNN, a fully-quantized spiking neural network with ultra-low precision weights, membrane potential, and high spatial-channel sparsity, enabling the end-to-end low precision with significantly reduced operations on SNN. First, we propose an integer-only quantization scheme for the membrane potential with a stacked surrogate gradient function, a simple-yet-effective method that enables the smooth learning process of quantized SNN training. Second, we implement spatial-channel pruning with membrane potential prior, toward reducing the layer-wise computational complexity, and floating-point operations (FLOPs) in SNNs. Finally, to further improve the accuracy of low-precision and sparse SNN, we propose a self-adaptive learnable potential threshold for SNN training. Equipped with high biological adaptiveness, minimal computations, and memory utilization, SpQuant-SNN achieves state-of-the-art performance across multiple SNN models for both event-based and static image datasets, including both image classification and object detection tasks. The proposed SpQuant-SNN achieved up to 13× memory reduction and >4.7× FLOPs reduction with ~1.8% accuracy degradation for both classification and object detection tasks, compared to the SOTA baseline.