The burning number $b(G)$ of a graph $G$, introduced by Bonato, is the minimum number of steps to burn the graph, which is a model for the spread of influence in social networks. In 2016, Bonato et al. studied the burning number of paths and cycles, and based on these results, they proposed a conjecture on the upper bound for the burning number. In this paper, we determine the exact value of the burning number of $Q$ graphs and confirm this conjecture for $Q$ graph. Following this, we characterize the single tail and double tails $Q$ graph in term of their burning number, respectively.