In the materials design domain, much of the data from materials calculations is stored in different heterogeneous databases with different data and access models. Therefore, integrating data from different sources is challenging. As ontology-based access and integration alleviates such issues, we developed the Materials Design Ontology (MDO) for increasing interoperability among computational materials databases. Additionally, we introduced a generic framework for using GraphQL in which a global domain ontology informs the generation of a GraphQL server that answers requests by querying heterogeneous data sources. By using MDO in this framework, we can generate a GraphQL server for data integration among heterogeneous computational materials databases.