In this talk, we will explain the existence of a universal braided compact quantum group acting on a graph $C^*$-algebra in the twisted monoidal category of $C^*$-algebras equipped with an action of the circle group. To achieve this we construct a braided version of the free unitary quantum group. Finally, we will compute this universal braided compact quantum group for the Cuntz algebra. This is a joint work in progress with Suvrajit Bhattacharjee and Soumalya Joardar.