I do not think it is possible to get a closed-form of either $f_n$ or $F_n$.

For brevity, let's rewrite your recurrence equation for the generating function $a_n:=F_n(x)$ as $a_n = q a_{n-1} + p a_{n-1}^2$ with the initial condition $a_0=x$. Then, change of variables $b_n = p a_n + \frac{q}{2}$ will reduce this equation to $b_n = \frac{(1+p)q}{4} + b_{n-1}^2$ with $b_0 = p x + \frac{q}{2}$, which is known as a quadratic map generating the Mandelbrot set and is not generally closed-form solvable. See [[1]], which gives more info on possible solutions; however, it will only get you so far that you can find $F_n(x)$ for some specific points $x$. I guess this if of little interest in your situation.

  [1]: http://mathworld.wolfram.com/QuadraticMap.html "Quadratic Map"