Fame's fleeting chariot

Is "Fame's fleeting chariot" a quotation from something
I ought to know?

Doesn't sound familiar to me. Are you perhaps thinking of "But at my back I always hear/time's winged chariot hurrying near..."?

This method often works: Go to www.google.com and type into the box a few words from the quotation you're trying to identify. I did so with "fame fleeting chariot," and the links that came up didn't produce the phrase "fame's fleeting chariot" in that order. The phrase is therefore probably not a literary quotation.