The Answer
No
Rolling Stone magazine was founded in 1967. By then The Rolling Stones were at the height of their fame, having been founded 5 years earlier in 1962.
Rolling Stone founder, editor and publisher Jann Wenner explained the naming philosophy himself in the “Letter from the Editor” in the very first Nov 1967 issue.
The name of it is Rolling Stone, which comes from an old saying: ‘A Rolling Stone gathers no moss.’ Muddy Waters used the name for a song he wrote; The Rolling Stones took their name from Muddy’s song, and “Like A Rolling Stone” was the title of Bob Dylan’s first rock and roll record.