I'm not familiar with the nonpositive case (negative since you exclude hyperkahler case) but there are no such examples known for postive quaternion Kahler manifolds (i.e. those with positive scalar curvature). They are all conjectured to be symmetric spaces (conjecture of LeBrunn and Salamon) and this conjecture has been verified in dimensions 4 (Hitchin) and and 8 (Poon and Salamon). Certainly, only symmetric examples are known such as $\mathbb HP^n, Gr_2(\mathbb C^{n+2})$, $\widetilde {Gr_4}(\mathbb R^{n+4})$ (the
Grassmanian of oriented real $4$-planes) and a few   exceptional spaces such as
$G_2/SO(4)$.