  Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class
of fully non-linear elliptic equations on certain compact hyperhermitian
manifolds. By adapting the approach of Sz\'{e}kelyhidi to the hypercomplex
setting, we prove some a priori estimates for solutions to such equations under
the assumption of existence of $\mathcal{C}$-subsolutions. In the estimate of
the quaternionic Laplacian, we need to further assume the existence of a flat
hyperk\"ahler metric. As an application of our results we prove that the
quaternionic analogue of the Hessian equation and Monge-Amp\`ere equation for
$(n-1)$-plurisubharmonic functions can always be solved on compact flat
hyperk\"ahler manifolds.
