NASA's Solar Dynamics Observatory (SDO) mission collects large data volumes of the Sun's daily activity. Data compression is crucial for space missions to reduce data storage and video bandwidth requirements by eliminating redundancies in the data. In this paper, we present a novel neural Transformer-based video compression approach specifically designed for the SDO images. Our primary objective is to efficiently exploit the temporal and spatial redundancies inherent in solar images to obtain a high compression ratio. Our proposed architecture benefits from a novel Transformer block called Fused Local-aware Window (FLaWin), which incorporates window-based self-attention modules and an efficient fused local-aware feed-forward (FLaFF) network. This architectural design allows us to simultaneously capture short-range and long-range information while facilitating the extraction of rich and diverse contextual representations. Moreover, this design choice results in reduced computational complexity. Experimental results demonstrate the significant contribution of the FLaWin Transformer block to the compression performance, outperforming conventional hand-engineered video codecs such as H.264 and H.265 in terms of rate-distortion trade-off.