Almost five years exactly after its conception, the American L.G.B.T.Q.+ Museum — which aims to open its doors in 2024 — has found its first executive director. The museum announced Tuesday that Ben Garcia, formerly the deputy executive director and chief learning officer of Ohio History Connection, would inaugurate the role in mid-February. At Ohio History Connection, a statewide history organization, Garcia managed more than 50 museums and historic sites, the state archives and the state historic preservation office. Before that, he worked as the deputy director at the Museum of Us in San Diego and the head of interpretation and operations at the Phoebe A. Hearst Museum of Anthropology in Berkeley, Calif.