For example, in the interval $[-1, 1]$, we have our nodes of the two-point rule as $1/\sqrt3$ and $-1/\sqrt3$. (edit: These two nodes were obtained because they are the roots of the third Legendre polynomial $x^2-1/3$ over the interval $[-1, 1]$.)
But are these same nodes used when we approximate the integral of any function in our interval?
It seems like it is the case, but I'm asking here because it feels unintuitive. There might be some functions for which there are better approximations when we choose different nodes.
Yes, the nodes will be the same for any function. This happens because the criteria being used for deciding if a quadrature rule is better then the next one is the maximum degree of polynomials for which it is exact, and this choice of nodes/weighs gives your best possible choice regarding that criteria.