If a Fourier series converges, will it converge to a periodic function? It seems logical since it is a trigonometric series. But often we are told to derive the Fourier series of functions like $x^2$, which are not periodic.
The sum of a Fourier series is periodic. As said in a comment, when people talk about the Fourier series of $x^2$, they actually mean the Fourier series of this function:
which is the result of restricting $x^2$ to the interval $[-\pi, \pi]$ (or whatever interval you use for Fourier series), and then extending that periodically.