Maybe a bit late, but anyway.

You know electric charges, and you probably heard about dipoles. A quadrupole is just the next step. In general, you can expand charge densities into multipoles. but a quadrupole is nothing else than an arrangement of 4 charges, with no net charge and no net dipole.

It turns out that nuclei with a spin larger than 1/2 have a quadrupole moment. The energy of a charge is given by it's product with the potential. the energy of a dipole is given by its product with the electric field (the derivative of the potential), and a quadrupole's energy is given by it's product with the electric field gradient (derivative of the electric field).

Now if your nucleus has a quadrupole and your molecule has an electric field gradient at the position of the nucleus, you get a quadrupole interaction.

it's a bit more complicated than that, because you measure spin interactions at the end. but the direction of the spin magnetic moment is connected to the quadrupole moment.

protons have I=1/2, no quadrupole. deuterium has I=1, with a quadrupole.

In which lab are you working, if I may ask?