Write $p=1-q$, $qN-j=(N-j)q-jp$, the $j$-th summand is $$(j(1-j+J)p+(j-J)(N-j)q)q^jp^{N-j} {N\choose j}.$$ Expand teh brackets and look it as a homogeneous polynomial in $p$ and $q$ of degree $N+1$. The coefficient of $q^{j+1}p^{N-j}$ equals $$ (j-J)(N-j){N\choose j}+(j+1)(J-j){N\choose j+1}=0. $$

Write $p=1-q$, $qN-j=(N-j)q-jp$, the $j$-th summand is $$(j(1-j+J)p+(j-J)(N-j)q)q^jp^{N-j} {N\choose j}.$$ Expand the brackets and consider it as a homogeneous polynomial in $p$ and $q$ of degree $N+1$. The coefficient of $q^{j+1}p^{N-j}$ equals $$ (j-J)(N-j){N\choose j}+(j+1)(J-j){N\choose j+1}=0. $$