Lithium zirconate (LZO) is a prototype material for studies of Li+ ion mobility with a wide range of possible applications such as a ceramic breeder for nuclear reactors, a reversible sorbent for carbon dioxide capture, a coating for cathodes and anodes or even directly as an anode material in lithium-ion batteries (LIBs). Solid state nuclear magnetic resonance (NMR) is a powerful experimental technique with the potential to provide microscopic insights into Li+ ion dynamics in solid materials, in particular if combined with theory to interpret the measured spectra. We use first-principles atomistic simulations based on density functional theory (DFT) to investigate the Li+ ion migration mechanisms in LZO. Computed barrier heights for several possible Li+ ion exchange pathways are in very good agreement with the experimentally reported values and confirm the relevance of lithium vacancies for the observed Li+ ion mobilities. Additionally, Li-7 NMR isotropic spectral parameters such as quadrupolar coupling constants and chemical shifts, can be obtained by the gauge-including projector-augmented-wave (GIPAW) method in very good agreement with the experimental values, underpinning the validity of the computational models. The close analysis of these spectral parameters shows a clear correlation to simple descriptors for the local structural environment of the Li+ ions, opening up a pathway to further modelling based on computationally cheaper methods. We note, however, that there is also a consistently poor agreement with experimental data for Li-7 anisotropic properties like the asymmetry parameter of the electric field gradient (EFG) tensor, which calls for further theoretical method development in this field.