Pedestrian trajectory prediction aims to forecast future movements based on historical paths. Spatial-temporal (ST) methods often separately model spatial interactions among pedestrians and temporal dependencies of individuals. They overlook the direct impacts of interactions among different pedestrians across various time steps (i.e., high-order cross-time interactions). This limits their ability to capture ST inter-dependencies and hinders prediction performance. To address these limitations, we propose UniEdge with three major designs. Firstly, we introduce a unified ST graph data structure that simplifies high-order cross-time interactions into first-order relationships, enabling the learning of ST inter-dependencies in a single step. This avoids the information loss caused by multi-step aggregation. Secondly, traditional GNNs focus on aggregating pedestrian node features, neglecting the propagation of implicit interaction patterns encoded in edge features. We propose the Edge-to-Edge-Node-to-Node Graph Convolution (E2E-N2N-GCN), a novel dual-graph network that jointly models explicit N2N social interactions among pedestrians and implicit E2E influence propagation across these interaction patterns. Finally, to overcome the limited receptive fields and challenges in capturing long-range dependencies of auto-regressive architectures, we introduce a transformer encoder-based predictor that enables global modeling of temporal correlation. UniEdge outperforms state-of-the-arts on multiple datasets, including ETH, UCY, and SDD.