Spatial autocorrelation statistic: Moran's I (or Mantel test) gives a measure of homogeneity/heterogeneity of diversity between nearby regions.
Note about the Modifiable Areal Unit Problem (MAUP): since the boundaries of grid squares are arbitrary and unrelated to ecological "zones", there is the possibility of division-scheme skewing the distribution of diversity in a systematic way. It is possible to test for this (MAUP sensitivity analysis) but this would mean regenerating the statistics using different division-schemes, a lengthy process. See https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4590437/ for a demonstration of the effects of boundary choice and spatial autocorrelation