We study the effects of interactions on low-energy density of states (SdH) and magnetization (dHvA) quantum oscillations (QOs) in a Kondo insulator. We consider a hybridization-gap insulator and incorporate interactions among N flavors of the heavy f electrons using a periodic Anderson model with Sachdev-Ye-Kitaev (SYK) interactions at each site. This model is exactly solvable in the large-N limit. We analytically and numerically show that the insulating state remains stable in terms of spectrum, thermodynamics and transport at low temperatures even for interaction strength much larger than non-interacting band gap. We derive extended Lifshitz-Kosevich (LK)-like analytical formulae, distinct for SdH and dHvA QOs, for the interacting insulator. We find that the SdH oscillation frequencies, F0 ± δF, corresponding to the conduction and valence band edges, can be weakly renormalized by interaction from their non-interacting values. In contrast, dHvA QOs are not renormalized by interactions, occurring at the frequency F0 for non-interacting unhybridized bands. The QO amplitudes show a remarkable difference: the dHvA amplitude is essentially unaffected by interaction, while the SdH amplitude is strongly enhanced