In electronic health record (EHR) mining, learning high-quality representations of medical concepts (e.g., standardized diagnosis, medication, and procedure codes) is fundamental for downstream clinical prediction. However, robust concept representation learning is hindered by two key challenges: (i) clinically important cross-type dependencies (e.g., diagnosis-medication and medication-procedure relations) are often missing or incomplete in existing ontology resources, limiting the ability to model complex EHR patterns; and (ii) rich clinical semantics are often missing from structured resources, and even when available as text, are difficult to integrate with KG structure for representation learning. To address these challenges, we present MedCo, an LLM-empowered graph learning framework for medical concept representation. MedCo first builds a global knowledge graph (KG) over medical codes by combining statistically reliable associations mined from EHRs with type-constrained LLM prompting to infer semantic relations. It then utilizes LLMs to enrich the KG into a text-attributed graph by generating node descriptions and edge rationales, providing semantic signals for both concepts and their relationships. Finally, MedCo jointly trains a LoRA-tuned LLaMA text encoder with a heterogeneous GNN, fusing text semantics and graph structure into unified concept embeddings. Extensive experiments on MIMIC-III and MIMIC-IV show that MedCo consistently improves prediction performance and serves as an effective plug-in concept encoder for standard EHR pipelines.