We compute the exact value of the quantum chromatic numbers of Hadamard graphs of order $n=2^N$ for $N$ a multiple of $4$ using the upper bound derived by Avis, Hasegawa, Kikuchi, and Sasaki, as well as an application of the Hoffman-like lower bound of Elphick and Wocjan that was generalized by Ganesan for quantum graphs. As opposed to prior computations for the lower bound, our approach uses Ito's results on conjugacy class graphs allowing us to also find bounds on the quantum chromatic numbers of products of Hadamard graphs. In particular, we also compute the exact quantum chromatic number of the categorical product of Hadamard graphs.