Molecule with a quadrupole moment in an electric field
To determine the force on an arbitrary multipole moment, we first expand the E-field in a Taylor-series around the point $\vec{r}=0$: 

$$\vec{E}(\vec{r})=\vec{E}_0+\vec{r}\cdot(\nabla\vec{E})_0+\frac{1}{2}\vec{r}\vec{r}:(\nabla\nabla\vec{E})_0+...$$

where the number of dots in the product denotes how many indices to contract over. We can then determine the force using this expansion, since it is just the charge multiplied by the E-field:

$$\vec{F}=q\vec{E}_0+\vec{\mu}\cdot(\nabla\vec{E})_0+\frac{1}{2}\mathbf{\Theta}:(\nabla\nabla\vec{E})_0+...$$
Typically, people use the traceless quadrupole, which would add an additional factor of $\frac{1}{3}$ in the third term, but I'm going to work with the basic definition throughout. So this shows that a quadrupole subject to a large electric field double gradient (not sure of the correct terminology for this) will experience a force even if it is charge neutral and has no net dipole.

For completeness, we can also write the energy since the force is just the gradient of the energy:

$$W=q\phi-\mu\cdot\vec{E}-\frac{1}{2}\mathbf{\Theta}:(\nabla\vec{E})+...$$
where we see, as TryHard said, that the E-field gradient interacts with the quadrupole.

As an aside, one can also obtain the torque (cross product of force and r) on an arbitrary multipole:
$$\vec{T}=\vec{\mu}\times\vec{E}_0+\frac{1}{2}\mathbf{\Theta}\dot{\times}(\nabla\vec{E})_0+...$$

(I'm using the nonstandard notation $A\dot{\times}B$ to mean $\sum_{ijkl}\epsilon_{jkl}e_iA_{jl}B_{lk}$, where $\epsilon_{jkl}$ is the Levi-Civita operator and the $e_i$ are a set of orthonormal unit vectors). This suggests that a quadrupole will experience a torque when interacting with a field gradient.

For more information on this, I would read chapter 3 of Jeanne McHale's Molecular Spectroscopy. There are some typos and notational quirks(e.g. in the equation for torque, McHale just writes a cross product for the 2nd term, which isn't defined between matrices), but its overall a good book to give background in classical electrostatics and a quantum mechanical description of spectroscopy.