A wealth of atomistic information is contained within a self-assembled quantum dot (QD), associated with its chemical composition and the growth history. In the presence of quadrupolar nuclei, as in InGaAs QDs, much of this is inherited to nuclear spins via the coupling between the strain within the polar lattice and the electric quadrupole moments of the nuclei. Here, we present a computational study of the recently introduced inverse spectra nuclear magnetic resonance technique to assess its suitability for extracting such structural information. We observe marked spectral differences between the compound InAs and alloy InGaAs QDs. These are linked to the local biaxial and shear strains, and the local bonding configurations. The cation-alloying plays a crucial role especially for the arsenic nuclei. The isotopic line profiles also largely differ among nuclear species: While the central transition of the gallium isotopes have a narrow linewidth, those of arsenic and indium are much broader and oppositely skewed with respect to each other. The statistical distributions of electric field gradient (EFG) parameters of the nuclei within the QD are analyzed. The consequences of various EFG axial orientation characteristics are discussed. Finally, the possibility of suppressing the first-order quadrupolar shifts is demonstrated by simply tilting the sample with respect to the static magnetic field. ; Comment: Published version, 17 pages, 18 figures