We initiate a systematic study of quantum properties of finite graphs, namely, quantum asymmetry, quantum symmetry, and quantum isomorphism. We define the Schmidt alternative for a class of graphs, which reveals to be a useful tool for studying quantum symmetries of graphs. After showing that quantum isomorphic graphs have quantum isomorphic centers and connected components, we solve the aforementioned problems for the classes of cographs and forests. We also compute their quantum automorphism groups for the first time. In doing so, we extend to the noncommutative setting a theorem of Jordan. Using general results on $\mathcal F$-cographs, we extend the precedent results to $\mathcal G_5$-cographs and tree-cographs, two distinct strictly proper superclasses of cographs and forests respectively. Finally, we show that quantum isomorphic planar graphs are isomorphic.