  We consider the evolution of stellar hierarchical quadruple systems in the
2+2 (two binaries orbiting each other's barycentre) and 3+1 (triple orbited by
a fourth star) configurations. In our simulations, we take into account the
effects of secular dynamical evolution, stellar evolution, tidal evolution and
encounters with passing stars. We focus on type Ia supernovae (SNe Ia) driven
by collisions of carbon-oxygen (CO) white dwarfs (WDs). Such collisions can
arise from several channels: (1) collisions due to extremely high
eccentricities induced by secular evolution, (2) collisions following a
dynamical instability of the system, and (3) collisions driven by semisecular
evolution. The systems considered here have initially wide inner orbits, with
initial semilatus recti larger than 12 AU, implying no interaction if the
orbits were isolated. However, taking into account dynamical evolution, we find
that $\approx 0.4$ ($\approx 0.6$) of 2+2 (3+1) systems interact. In
particular, Roche Lobe overflow can be triggered possibly in highly eccentric
orbits, dynamical instability can ensue due to mass-loss-driven orbital
expansion or secular evolution, or a semisecular regime can be entered. We
compute the delay-time distributions (DTDs) of collision-induced SNe Ia, and
find that they are flatter compared to the observed DTD. Moreover, our combined
SNe Ia rates are $(3.7\pm0.7) \times 10^{-6} \,\mathrm{M}_\odot^{-1}$ and
$(1.3\pm0.2) \times 10^{-6} \,\mathrm{M}_\odot^{-1}$ for 2+2 and 3+1 systems,
respectively, three orders of magnitude lower compared to the observed rate, of
order $10^{-3} \, \mathrm{M}_\odot^{-1}$. The low rates can be ascribed to
interactions before the stars evolve to CO WDs. However, our results are lower
limits given that we considered a subset of quadruple systems.
