The shooting occurred at about 11 a.m. in the 800 block of South Third Street, Sgt. Jason Dwyer said.
It was not known if the second victim's injuries are life-threatening, Dwyer said.
The shooting occurred shortly after a deputy-involved shooting about a mile and a half away near the intersection of South Montgomery Street and Park Avenue, Dwyer said.
San Jose police are investigating the incident, which involved deputies from the Santa Clara County Sheriff's Office but took place within the Police Department's jurisdiction. According to Dwyer, those deputies fired at a suspect who was later taken into custody.
A sheriff's spokesman said that the agency was not releasing information about the shooting.