No. His father, Valentine Fleming, was born a year after Alexander Fleming was. Plus, Ian is of British nationality while Alexander is of Scottish nationality. However, Valentine's father was named Robert Fleming, and surprisingly enough Alexander named his son Robert also.
Alexander Fleming affected the world by accidentally inventing penicillin, the world's first commercial antibiotic. There are theories that penicillin was invented prior to Alexander Fleming, but most historians accredit the discovery to him.