You should take the derivative with respect to $x$ of both sides. Then you obtain the differential equation $$ f'(x) = \frac{g(x) f(x)}{\sqrt{h(x)^2+f(x)^2}}$$ and $f(0)=0$. The latter can be solved using

g[x_] := x^2;
h[x_] := x^2/2 + c;
DSolve[{f'[x] == g[x] f[x]/Sqrt[h[x]^2 + f[x]^2], f[0]==0}, f[x], x]

It turns out that Mathematica is not able to solve the resulting differential equation.

Looking at the numerical solution (setting a value for $c$ and replacing DSolve by NDSolve), I believe that the only solution to your problem is $$ f(x) \equiv 0.$$

You should take the derivative with respect to $x$ of both sides. Then you obtain the differential equation $$ f'(x) = \frac{g(x) f(x)}{\sqrt{h(x)^2+f(x)^2}}$$ and $f(0)=0$. The latter can be solved using

g[x_] := x^2;
h[x_] := x^2/2 + c;
DSolve[{f'[x] == g[x] f[x]/Sqrt[h[x]^2 + f[x]^2], f[0]==0}, f[x], x]

It turns out that Mathematica is not able to solve the resulting differential equation.

Looking at the numerical solution (setting a value for $c$ and replacing DSolve by NDSolve), I believe that the (only) solution to your problem is $$ f(x) \equiv 0.$$