In this paper, we investigate uniqueness problems of certain types of $q$-difference polynomials, which improve some results in [20]. However, our proof is different from that in [20]. Moreover, we obtain a uniqueness result in the case where $q$-differences of two entire functions share values as well. This research also shows that there exist two sets, such that for a zero-order non-constant meromorphic function $f$ and a non-zero complex constant $q$, $E(S_j,f)=E(S_j,{Delta}_qf)$ for $j=1,2$ imply $f(z)=t{Delta}_qf$, where $t^n=1$. This gives a partial answer to a question of Gross concerning a zero order meromorphic function $f(z)$ and $t{Delta}_qf$.