Matter composed of hyperons has been hypothesized to occur in neutron stars at densities slightly above the nuclear saturation density and in many descriptions gives rise to a significant softening in the equation of state (EoS). This softening would be at odds with the constraints from neutron star observations and ab initio nuclear matter computations at low density. This inconsistency is known as the hyperon puzzle. We show that Quarkyonic Matter models, which take into account the quark substructure of baryons, can mitigate the hyperon puzzle. We demonstrate two important consequences of the quark substructure effects. First, the hyperon threshold is shifted to a higher density as neutrons preoccupy the phase space for down quarks, preventing the emergence of hyperons at low energy. Secondly, the softening in the EoS becomes mild even above the hyperon threshold density because only little phase space is available for low-energy hyperons; increasing hyperon density quickly drives hyperons into the relativistic regime. In this work, we illustrate these two effects for a matter composed of charge-neutral baryons, using the ideal dual Quarkyonic (IdylliQ) model for three flavors. This model incorporates the Quarkyonic duality and allows us to manifestly express the quark Pauli blocking constraints in terms of the baryon occupation probability. The extension to neutron star matter is also briefly discussed.