Real-life graphs from various application domains like social networks, transportation networks, and citation networks evolve over time. These evolving graphs can be modeled as (i) interactions between two nodes in a graph and (ii) interactions associated with a single node. Deep learning techniques using Graph Neural Networks (GNNs) are used for analyzing spatial and temporal properties of graphsfrom these application domains. Analyzing temporal graphs is challenging in comparison to static graphs, hence warranting the need for a GNN variant named Temporal Graph Neural Networks (TGNNs). We propose STGraph, a framework to program TGNNs. The proposed framework extends Seastar, a vertex-centric programming model for training static GNNs on GPUs. STGraph supports TGNNs for static temporal and discrete-time dynamic graphs (DTDGs). Existing TGNN frameworks store DTDGs as separate snapshots, incurring high memory overhead. As an improvement, STGraph constructs each snapshot on demand during training. This is achieved by integrating the system with dynamic graph data structures capable of building graph snapshots from temporal updates. Additionally, we present improvements to the Seastar design for easier maintenance and greater software portability. STGraph is benchmarked against Pytorch Geometric Temporal (PyG-T) on an NVIDIA GPU. For static-temporal graphs, STGraph shows up to 1.22× speedup and up to 2.14× memory improvement over PyG-T. For DTDGs, STGraph exhibits up to 1.70× speedup and 1.52× memory improvement over PyG-T.