Knowledge graph completion (KGC) aims to identify missing triples in a knowledge graph (KG). This is typically achieved through tasks such as link prediction and instance completion. However, these methods often focus on either static knowledge graphs (SKGs) or temporal knowledge graphs (TKGs), addressing only within-scope triples. This paper introduces a new generative completion framework called Generative Subgraph-based KGC (GS-KGC). GS-KGC employs a question-answering format to directly generate target entities, addressing the challenge of questions having multiple possible answers. We propose a strategy that extracts subgraphs centered on entities and relationships within the KG, from which negative samples and neighborhood information are separately obtained to address the one-to-many problem. Our method generates negative samples using known facts to facilitate the discovery of new information. Furthermore, we collect and refine neighborhood path data of known entities, providing contextual information to enhance reasoning in large language models (LLMs). Our experiments evaluated the proposed method on four SKGs and two TKGs, achieving state-of-the-art Hits@1 metrics on five datasets. Analysis of the results shows that GS-KGC can discover new triples within existing KGs and generate new facts beyond the closed KG, effectively bridging the gap between closed-world and open-world KGC.