He would be 43 in 1940. Quentin was born in 1897.
Now a Roosevelt vs. Roosevelt race would be interesting. But, then again, Quentin might not take the risk. As I recall, the Roosevelts didn't like losing, and weren't apt to run in nearly unwinable races.
So perhaps he waits for FDR to retire, assuming his survival doesn't butterfly FDR's terms away.
I think Quentin probably has to pay his dues somehow before he can sit in the Governors chair, probably works on some campaigns, serves in congress. His chance for the Governors seat would come in 1938, he defeats Thomas Dewey in the primary, and wins the election. (Dewey only lost by a percentage point, and Quentin has the Roosevelt name to back him.)
Now, again if Quentin doesn't think he can beat Franklin, he won't run against him.
Same goes in 1944, assuming the war doesn't go differently, and FDR decides not to run again.
Now in 1948, Quentin's been Governor for a decade. And this would be an excellent chance to have him become President.
Truman running against a man named Roosevelt.
I'd hate to be Truman in that scenario.