While modern times may have us using this phrase as a substitute for caring (i.e. I don't give a flying fuck what you do, just get me a drink) the true origins aren't quite the same.
The expression dates back to the 1845 in a volume by Thomas Rowlandson entitled: "Pretty Little Games for Young Ladies and Gentleman: With Pictures of Good Old English Sports and Pastimes." Contained within the volume were ten etchings each along side a poem that appears to also be the work of Mr. Rowlandson. The volume celebrated such past-times as fellatio, cunnilingus and others.
In this particular etching
a young man and his lady are on
horseback at a
gallop. Her skirt is raised to show her rear, and his mighty
penis is pushed beneath as to enter her
vagina. Both faces are
flushed with passion and are accompanied by the following
verse:
Well mounted on a mettled steed
Famed for his strength as well as speed
Corinna and her favorite buck
Are pleas’d to have a flying fuck.
So think of that the next time you announce that you give a flying fuck. Somebody might just take you seriously.
References:
Playboy’s Book of Forbidden Words p. 119
Robert A. Wilson, ed.
Playboy Press, 1972. 427 Wi69p
Etymology of a Dirty Word: http://www.urbanlegends.com/language/etymology/fuck/fuck_references.html
Etymology: http://www.geocities.com/etymonline/f3etym.htm