The non-Euclidean geometry inherent in graph structures fundamentally impedes cross-graph knowledge transfer. Drawing inspiration from texture transfer in computer vision, we pioneer topological primitives as transferable semantic units for graph structural knowledge. To address three critical barriers - the absence of specialized benchmarks, aligned semantic representations, and systematic transfer methodologies - we present G²SN-Transfer, a unified framework comprising: (i) TopoGraph-Mapping that transforms non-Euclidean graphs into transferable sequences via topological primitive distribution dictionaries; (ii) G²SN, a dual-stream architecture learning text-topology aligned representations through contrastive alignment; and (iii) AdaCross-Transfer, a data-adaptive knowledge transfer mechanism leveraging cross-attention for both full-parameter and parameter-frozen scenarios. Particularly, G²SN is a dual-stream sequence network driven by ordinary differential equations, and our theoretical analysis establishes the convergence guarantee of G²SN. We construct STA-18, the first large-scale benchmark with aligned topological primitive-text pairs across 18 diverse graph datasets. Comprehensive evaluations demonstrate that G²SN achieves state-of-the-art performance on four structural learning tasks (average 3.2\% F1-score improvement), while our transfer method yields consistent enhancements across 13 downstream tasks (5.2\% average gains) including 10 large-scale graph datasets. The datasets and code are available at https://anonymous.4open.science/r/UGSKT-C10E/.