Connecting two points on the boundary of an open, connected and bounded set $Q\subset \mathbb{R}^2$ with a path $\gamma\subset \overline{Q}$

Question

Let $Q$ be an open connected and bounded subset of $\mathbb{R}^2$. Is it true that for any two points $x,y\in \partial Q$, there is a path $\gamma \subset \overline{Q}$ which connects $x$ and $y$? I know that $\overline{Q}$ is connected and $Q$ is path-connected, but I still wasn't able to answer my question.