Giants rookie outfielder John Bowker has homered in each of his first two major league games, the first San Francisco Giant player to do so. There could be a New York Giant player, but I can't check back to that era.
The last major leaguer to start his career with home runs in his first two games was Elijah Dukes of Tampa Bay last season on April 2 and April 5, 2007. The last NL player to do it was Keith McDonald of the Cardinals on July 4 and July 6, 2000. McDonald's homers came in his first two major league at bats.