The answer is *yes*. In fact, there is the following

>**Proposition.** Every $2$-dimensional rational singularity $(X, \, x)$ is analytically $\mathbb{Q}$-factorial, i.e. there exists an analytic neighborhood $V$ of $x$ such that every Weil divisor on $V$ is a $\mathbb{Q}$-Cartier divisor. 

>In particular, every $2$-dimensional rational singularity is $\mathbb{Q}$-Gorenstein.

See S. Ishii, *[Introduction to Singularities][1]*, Theorem 7.3.2.


  [1]: http://www.springer.com/us/book/9784431550808