Mechanical properties are fundamental indicators of the structural stability and functional reliability of materials, as they determine whether a material can sustain mechanical loads while maintaining its intended performance. Among various material classes, two-dimensional (2D) materials, characterized by atomically thin layered structures, exhibit mechanical responses that are highly sensitive to chemical composition and crystal structures. However, obtaining stiffness tensors for 2D materials remains computationally demanding, even after acquiring relaxed structures through geometry optimization, because elastic constants require evaluating second-order derivatives of the total energy with respect to applied strain within density functional theory calculations. In this work, a large language model (LLM)-based framework is developed to predict stiffness tensors from crystal structures of 2D materials by jointly leveraging graph-based representations and user-provided textual prompts. The crystal structure is encoded as a graph to capture topological connectivity and interaction-level information, while textual prompts provide flexible and human-interpretable descriptions of the prediction task. By integrating graph embeddings with text embeddings within the LLM, the proposed framework provides a unified representation that aligns potentially ambiguous user instructions with physically meaningful structural information. Hand-crafted descriptors and task-specific regression models remain common in current materials property prediction workflows, often requiring substantial domain expertise and restricting the flexibility of input representations. The proposed method overcomes these limitations by leveraging the internal knowledge accumulated in large language models together with structured graph information, enabling users with limited domain expertise to perform mechanical property predictions across diverse material systems with reduced reliance on prior heuristic judgment. Moreover, the same architecture can be naturally extended beyond mechanical properties to predict chemical or functional responses, thereby supporting efficient multi-property evaluation. Overall, the proposed framework provides a practical approach for stiffness tensor prediction and material screening by combining graph-based structural representations with large language models.