  In this article, we give a first example of a pair of quantum isomorphic,
non-isomorphic strongly regular graphs, that is, non-isomorphic strongly
regular graphs having the same homomorphism counts from all planar graphs. The
pair consists of the orthogonality graph of the $120$ lines spanned by the
$E_8$ root system and a rank $4$ graph whose complement was first discovered by
Brouwer, Ivanov and Klin. Both graphs are strongly regular with parameters
$(120, 63, 30, 36)$. Using Godsil-McKay switching, we obtain more quantum
isomorphic, non-isomorphic strongly regular graphs with the same parameters.
