Talk about an expensive lunch date ... 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. For the record, 40-year-old A-Rod has reportedly made $342 million playing baseball. 42-year-old Anne (who runs a medical research company) reportedly raked in several billion dollars when she finalized her divorce from Brin in 2015. So, the question ... who paid for lunch? Wow!!! A-Roid looks as arrogant and disinterested in his date as he always does. Is he gay?