Existing low-light video enhancement methods are dominated by Convolution Neural Networks (CNNs) that are trained in a supervised manner. Due to the difficulty of collecting paired dynamic low/normal-light videos in real-world scenes, they are usually trained on synthetic, static, and uniform motion videos, which undermines their generalization to real-world scenes. Additionally, these methods typically suffer from temporal inconsistency (e.g., flickering artifacts and motion blurs) when handling large-scale motions since the local perception property of CNNs limits them to model long-range dependencies in both spatial and temporal domains. To address these problems, we propose the first unsupervised method for low-light video enhancement to our best knowledge, named LightenFormer, which models long-range intra- and inter-frame dependencies with a spatial-temporal co-attention transformer to enhance brightness while maintaining temporal consistency. Specifically, an effective but lightweight S-curve Estimation Network (SCENet) is first proposed to estimate pixel-wise S-shaped non-linear curves (S-curves) to adaptively adjust the dynamic range of an input video. Next, to model the temporal consistency of the video, we present a Spatial-Temporal Refinement Network (STRNet) to refine the enhanced video. The core module of STRNet is a novel Spatial-Temporal Co-attention Transformer (STCAT), which exploits multi-scale self- and cross-attention interactions to capture long-range correlations in both spatial and temporal domains among frames for implicit motion estimation. To achieve unsupervised training, we further propose two non-reference loss functions based on the invertibility of the S-curve and the noise independence among frames. Extensive experiments on the SDSD and LLIV-Phone datasets demonstrate that our LightenFormer outperforms state-of-the-art methods.