  Measurements of the $^{17}$O nuclear magnetic resonance (NMR) quadrupolar
spectrum of apical oxygen in HgBa$_{2}$CuO$_{4+\delta}$ were performed over a
range of magnetic fields from 6.4 to 30\,T in the superconducting state. Oxygen
isotope exchanged single crystals were investigated with doping corresponding
to superconducting transition temperatures from 74\,K underdoped, to 78\,K
overdoped. The apical oxygen site was chosen since its NMR spectrum has narrow
quadrupolar satellites that are well separated from any other resonance.
Non-vortex contributions to the spectra can be deconvolved in the time domain
to determine the local magnetic field distribution from the vortices. Numerical
analysis using Brandt's Ginzburg-Landau theory was used to find structural
parameters of the vortex lattice, penetration depth, and coherence length as a
function of magnetic field in the vortex solid phase. From this analysis we
report a vortex structural transition near 15\,T from an oblique lattice with
an opening angle of $73^{\circ}$ at low magnetic fields to a triangular lattice
with $60^{\circ}$ stabilized at high field. The temperature for onset of vortex
dynamics has been identified with vortex lattice melting. This is independent
of the magnetic field at sufficiently high magnetic field similar to that
reported for YBa$_2$Cu$_3$O$_7$ and Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$
and is correlated with mass anisotropy of the material. This behavior is
accounted for theoretically only in the limit of very high anisotropy.
