On the season, Greene is 3-7 with a 5.10 ERA in 12 starts.
His ERA was 6.00 in April, 5.18 in May and 2.87 so far in June.
He told MLB.com that being named player of the week is “a really cool accolade.”
“To be able to hit that in your first year is really, really special,” he said. “I was with my family actually when I saw it. They were proud of me. To experience that moment with them was cool.
“Still a lot of work to be done, but I definitely want to smell the roses and not be too caught up in trying to be perfect every time, but to enjoy it. Really cool.”