Here we present a new particle-mesh galactic N-body code that uses the full multigrid algorithm for solving the modified Poisson equation of the quasi-linear formulation of modified Newtonian dynamics (QUMOND). A novel approach for handling the boundary conditions using a refinement strategy is implemented and the accuracy of the code is compared with analytical solutions of Kuzmin discs. We then employ the code to compute the predicted rotation curves for a sample of five spiral galaxies from the THINGS sample. We generated static N-body realizations of the galaxies according to their stellar and gaseous surface densities and allowed their distances, mass-to-light ratios (M/L values) and both the stellar and gas scale-heights to vary in order to estimate the best-fitting parameters. We found that NGC 3621, NGC 3521 and DDO 154 are well fitted by MOND using expected values of the distance and M/L. NGC 2403 required a moderately larger M/L than expected and NGC 2903 required a substantially larger value. The surprising result was that the scale-height of the dominant baryonic component was well constrained by the rotation curves: the gas scale-height for DDO 154 and the stellar scale-height for the others. In fact, if the suggested stellar scale-height (one-fifth the stellar scale-length) was used in the case of NGC 3621 and NGC 3521 it would not be possible to produce a good fit to the inner rotation curve. For each of the four stellar dominated galaxies, we calculated the vertical velocity dispersions which we found to be, on the whole, quite typical compared with observed stellar vertical velocity dispersions of face-on spirals. We conclude that modelling the gas scale-heights of the gas-rich dwarf spiral galaxies will be vital in order to make precise conclusions about MOND.