  By characterizing all orthogonal polynomials sequences $(P_n)_{n\geq 0}$ for
which $$ (ax+b)(\triangle +2\,\mathrm{I})P_n(x(s-1/2))=(a_n x+b_n)P_n(x)+c_n
P_{n-1}(x),\quad n=0,1,2,\dots, $$ where $\,\mathrm{I}$ is the identity
operator, $x$ defines a $q$-quadratic lattice, $\triangle f(s)=f(s+1)-f(s)$,
and $(a_n)_{n\geq0}$, $(b_n)_{n\geq0}$ and $(c_n)_{n\geq0}$ are sequences of
complex numbers, we derive some new structure relations for some specific
families of orthogonal polynomials.
