Indetermined moment problems related to $q$-functional equations

For $q\in (0,1)$, $p_1,p_2,p\in \mathbb{R}_+$, we characterize all the solutions of the $q$-functional equations $(1+p_2q^{1/2}x)f(qx)=q^{\beta-1/2}(x+p_1q^{-1/2})f(x)$ and $f(qx)=q^{\beta- 1}(x^2+p^2q^{-1})f(x)$, $x>0$, $\beta\in \mathbb{R}$, and we also show that these solutions solve corresponding indetermined moment problems.