That's really the easiest part. We use Stardog, but there are a bunch of other databases you can use including Oracle Spatial and Graph, Fuseki, GraphDB, Virtu

That's really the easiest part. We use Stardog, but there are a bunch of other databases you can use including Oracle Spatial and Graph, Fuseki, GraphDB, Virtuoso, AllegroGraph, MarkLogic and Blazegraph. Many of these database will answer queries in json-ld, but since you are likely to want to have a backend anyway you can use Apache Jena or Eclipse RDF4J (previously Sesame) with Java to connect and extract data. Both Jena and RDF4J will let you output that data as JSON-LD.