The Earthquakes became the first visiting team to earn a point at The Home Depot Center, the Galaxy's home field.
Walker's tally in the 63rd minute canceled a first-half strike by Sasha Victorine and gave San Jose sole possession of first place in the Western Conference.
The Galaxy christened their new stadium with a pair of wins following a season-opening 0-4-4 road trip.
For the Quakes, Brian Mullan spun away from a defender and slipped the ball to Ian Russell running behind the defense. Russell drove a low shot across the face of the net, where Walker knocked it over the line for his first MLS goal.
Victorine scored San Jose's lone goal.