According to the book Spencer Tracy Fox Film Actor: The Pre-Code Legacy of a Hollywood Legend, Tracy arrived in Hollywood in the summer of 1930 to make Up the River. He had taken six weeks' leave from a play in New York, of which he was obligated to appear for several more months.
Somewhere else I read that Tracy left New York for California in June 1930 to make Up the River, but I can't remember the source.
It doesn't pinpoint the production date but does narrow it down based on an October 1930 premiere.