  In this paper we study queen's graphs, which encode the moves by a queen on
an $n\times m$ chess board, through the lens of chip-firing games. We prove
that their gonality is equal to $nm$ minus the independence number of the
graph, and give a one-to-one correspondence between maximum independent sets
and classes of positive rank divisors achieving gonality. We also prove an
identical result for toroidal queen's graphs.
