Davis won three Gold Gloves, and probably should have had more. Baseball-Reference has him as compiling 10.9 WAR of his 54.5 career WAR from his defense alone. FanGraphs rates his defense higher than his offensive WAR contributions, due to using TotalZone marks that rate him extremely high. There were no UZR stats in those days (obviously) and Fielding Bible doesn't go back that far. But it's pretty well established that Davis was one of the best defensive center fielders in Dodgers history.
Davis played for the LA Dodgers from 1960-1973. He had one more good season for the Montreal Expos in 1974, before just sort of hanging on with the Texas Rangers and St. Louis Cardinals the next season. He got into 141 games with the Padres in 1976, bounced around the minors for awhile and got into 43 games for the California Angels in 1979 for his last major league action.
I wouldn't say that Davis was a Hall of Fame player, but he was extremely good. He definitely should have won more Gold Gloves than he did. Sadly, Willie is no longer with us as he passed away at age 69 in March of 2010. You will be missed, Willie. Thanks for the memories.