It's called Mission San José, but this mission isn't in San Jose. It sits on the slopes above the Fremont plain, in the district of Fremont that still carries its name.
Founding
Mission San José was founded on June 11, 1797, the 14th mission in the California chain. The California Missions Foundation says Father Lasuén founded it and named it for Saint Joseph. Wikipedia says the site had been home for countless generations to people who spoke the San Francisco Bay Ohlone language. Many of the first residents had been baptized at Mission Santa Clara, 13 miles south. In 1825 Delta Yokuts was the main language in a mission community of 1,796 people.
Music and revolt
The foundation says Father Narciso Durán spent 27 years at the mission, where he organized a choir and a 30-piece orchestra that became famous across California. The mission's history wasn't peaceful. In 1828 Estanislao led a major uprising, and several military expeditions were sent to put it down. After Mexico secularized the missions in 1833, Mission San José went into a long decline.
Earthquake and rebuilding
The foundation says the 1868 earthquake nearly destroyed the mission. The monastery wing, which now houses the mission museum, was one of the few buildings to survive. A wooden Gothic church stood on the site for more than a century afterward and was then moved to Burlingame. In its place stands an authentic adobe replica of the church, built in 1985. The old church is still in use as a chapel of Saint Joseph Catholic Church, part of the Diocese of Oakland. Wikipedia says the site also has a visitor center and museum.