No. Gauss' work was not published until 1866, as a part of his collected works. Prior to that, there were various authors who published related algorithms (e.g. a paper by Everett in 1860, one published by Archibald Smith in 1846, and one published by F. Carlini in 1828, although these works only described restricted cases).
What does seem to be true is that Gauss was the first *recorded* discoverer of an FFT. He was also (apparently) the only author until Cooley & Tukey in 1965 to describe a general mixed-radix algorithm for any composite size.
(See the excellent paper, "Gauss and the History of the Fast Fourier Transform," by Heideman et al., IEEE ASSP Magazine, p. 14, October
1984.)
You're being a bit too pedantic for my taste; by "publish" in science, we usually mean "initiate the publication process".
Regards, Steven G. Johnson