  While excitonic instabilities in multiorbital systems recently have come
under scrutiny in a variety of transition-metal compounds, understanding
emergence of these instabilities from strong electronic interactions has
remained a challenge. Here, we present a sign-problem-free determinant quantum
Monte Carlo study of excitonic density orders in a half-filled two-orbital
Hubbard-Kanamori model with broken orbital degeneracy, which accounts for the
role of Hund's coupling in transition-metal compounds. For strong inverted
(negative) Hund's exchange, we find numerical evidence for the emergence of
excitonic density order, with competition between anti-ferro-orbital order and
$\mathbf{Q} = (\pi,\pi)$ excitonic density order as a function of orbital
splitting and Hund's coupling. While inverted Hund's coupling stabilizes a
spin-singlet excitonic density phase for weak orbital splitting, positive
Hund's coupling favors a spin-triplet excitonic density phase.
