We introduce and study quantized versions of Cop and Robber game. We achieve this by using graph-preserving quantum operations, which are the quantum analogues of stochastic operations preserving the graph. We provide the tight bound for the number of operations required to reach the given state. By extending them to the controlled operations, we define a quantum-controlled Cop and Robber game, which expands the classical Cop and Robber game, as well as the classically controlled quantum Cop and Robber game. In contrast to the typical scheme for introducing quantum games, we assume that both parties can utilise full information about the opponent's strategy. We show that the utilisation of the full knowledge about the opponent's state does not provide the advantage. Moreover, the chances of catching the Robber decrease for classical cop-win graphs. This result does not depend on the chosen model of evolution. On the other hand, the possibility to execute controlled quantum operations allows catching the Robber on almost all classical cop-win graphs. By this, we demonstrate that it is necessary to enrich the structure of correlations between the players' systems to provide a non-trivial quantized Cop and Robber game. Thus the quantum controlled operations offer a significant advantage over the classically controlled quantum operations.