We initiate the study of diameter computation in geometric intersection graphs from the fine-grained complexity perspective. A geometric intersection graph is a graph whose vertices correspond to some shapes in d-dimensional Euclidean space, such as balls, segments, or hypercubes, and whose edges correspond to pairs of intersecting shapes. The diameter of a graph is the largest distance realized by a pair of vertices in the graph. Computing the diameter in near-quadratic time is possible in several classes of intersection graphs [Chan and Skrepetos 2019], but it is not at all clear if these algorithms are optimal, especially since in the related class of planar graphs the diameter can be computed in O^e(n⁵/³) time [Cabello 2019, Gawrychowski et al. 2021]. In this work we (conditionally) rule out sub-quadratic algorithms in several classes of intersection graphs, i.e., algorithms of running time O(n^2-d) for some d > 0. In particular, there are no sub-quadratic algorithms already for fat objects in small dimensions: unit balls in R³ or congruent equilateral triangles in R². For unit segments and congruent equilateral triangles, we can even rule out strong sub-quadratic approximations already in R². It seems that the hardness of approximation may also depend on dimensionality: for axis-parallel unit hypercubes in R¹², distinguishing between diameter 2 and 3 needs quadratic time (ruling out (3/2-e)- approximations), whereas for axis-parallel unit squares, we give an algorithm that distinguishes between diameter 2 and 3 in near-linear time. Note that many of our lower bounds match the best known algorithms up to sub-polynomial factors. Ultimately, this fine-grained perspective may enable us to determine for which shapes we can have efficient algorithms and approximation schemes for diameter computation.