Here are the first pics of Alex Rodriguez out in public with his new girlfriend Anne Wojcicki ... the ex-wife of Google's billionaire co-founder, Sergey Brin.
The two had a late lunch together on Sunday at Green Lemon -- an awesome Mexican restaurant in Tampa, FL ... right near where the Yankees hold spring training.
We're told the two were there for more than an hour -- and were holding hands and acting couply the entire time. You can see in the pics Anne had her arm around A-Rod while the 2 chowed down.