We present a theoretical analysis of the electronic structure of GaN quantum dots (QD) with an adjacent threading dislocation. The QD carrier spectra and wavefunctions are calculated using a plane-wave expansion method within an 8-band k.p model. The method is very efficient, because the strain and built-in electric fields can be included through their discrete Fourier transforms. The QD structures considered have been analysed experimentally by other groups. The GaN QDs are truncated hexagonal pyramids on a wetting layer with an edge dislocation adjacent to each dot. The built-in piezoelectric potential strongly influences the localisation of the carrier wavefunctions. This potential pushes the electrons to the top of the dot, the holes to the bottom and, additionally, causes strong lateral confinement of the carriers. The effect of the dislocation strain field at the dot edge on the carrier states in each GaN/AlN QD is shown to be insignificant. Results are presented for the confined state energies and optical matrix elements for a range of different sized dots with and without dislocations. The size of the dot influences the energies and overlaps, but the presence of the dislocation has minimal effect. The dependence of the ground state optical transition energy on the size of the dot is in good agreement with experimental data.