We show for any n-gon. Suppose its vertices are the vectors $v_1,...,v_n$ with origin at the center of the circle. Then all of them have unit length. Now the sum of the squares of all the distances is $\frac12\sum\sum (v_i-v_j)^2$. Note that $(v_i-v_j)^2=v_i^2+v_j^2-2v_i\cdot v_j=2-2v_i\cdot v_j$. So the sum becomes $n^2-\left(\sum v_i\right)^2\leq n^2$, and equality is always possible by regular n-gon. So the maximum QM is $\sqrt{\frac{n}{n-1}}$.