High-fidelity compression of multispectral solar imagery remains challenging for space missions, where limited bandwidth must be balanced against preserving fine spectral and spatial details. We present a learned image compression framework tailored to solar observations, leveraging two complementary modules: (1) the Inter-Spectral Windowed Graph Embedding (iSWGE), which explicitly models inter-band relationships by representing spectral channels as graph nodes with learned edge features; and (2) the Windowed Spatial Graph Attention and Convolutional Block Attention (WSGA-C), which combines sparse graph attention with convolutional attention to reduce spatial redundancy and emphasize fine-scale structures. Evaluations on the SDOML dataset across six extreme ultraviolet (EUV) channels show that our approach achieves a 20.15%reduction in Mean Spectral Information Divergence (MSID), up to 1.09% PSNR improvement, and a 1.62% log transformed MS-SSIM gain over strong learned baselines, delivering sharper and spectrally faithful reconstructions at comparable bits-per-pixel rates. The code is publicly available at https://github.com/agyat4/sgraph .