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From the equation for the EFG tensor taken in its PAS, we can directly see that it has an axial symmetry by inspecting the components of the 3×3 matrix. The first two terms on the diagonal are identical, which means that if we compute the asymmetry parameter η we would get zero which means that we are in the axially symmetric case as seen in the lecture.
From the picture of the toy model, I believe we can see that we have axial symmetry because we can rotate the dumbbell around the z-axis but this would leave the electric field gradient unchanged since the distances between the positive and negative charges remains the same.
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