While large language models have revolutionized Text-to-SQL tasks, translating natural language into Graph Query Languages (Text2GQL) remains underexplored due to the topological heterogeneity and syntactic diversity of graph query languages (e.g., Cypher, Gremlin, SPARQL). Existing approaches often struggle with structural hallucinations and lack adaptability in cold-start scenarios. In this paper, we present a unified, training-free Text2GQL framework. First, Structural Twig Linking elevates schema grounding to the identification of semantic substructures (“twigs”), providing robust topological priors. Second, addressing data scarcity, Evolutionary In-Context Learning operates in a Tabula Rasa setting to implicitly construct a self-growing repository of verified examples driven by syntactic utility. Finally, our Adversarial Execution-Guided Correction agent enforces fidelity through synergistic static critique and dynamic verification. Experiments demonstrate significant improvements over baselines in both accuracy and executability across diverse GQLs. The code is available at https://github.com/nf202/Text2Graph.