Alex Rodriguez: 1st Pics with Billionaire Girlfriend ... Google Her

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.