Suppose $\Omega= (0,1)\times(0,1)\subset \mathbb R^2$. Assume that $f, g \in C^{\infty}(\Omega)$ and that $$ \int_\Omega \left(f(x_1,x_2)- \frac{m}{(n+1)}g(x_1,x_2)\right) x_1^n \,x_2^m \,dx_1\,dx_2 = 0 \quad \text{for all $n,m=0,1,\ldots$}.$$

Does it follow that $f \equiv g \equiv 0$ on $\Omega$?