I am not sure exactly what (or why) is requested for this one, but I offer a solution fitting one interpretation.

The first step is to convert the 36 squares of the first grid to the second grid for the sums (from 16 to 129) of the neighbors of the same coordinates in grid one. This is just busywork.

Then, apparently, we note which cells on the second grid have "base 10 squares". Five of the 36 cells meet that condition, giving 16, 36, 49, 64, and 100. Then we are to express each of these in a base greater than 10. For this:

16 (base 10) = 14 (base 12)

36 (base 10) = 33 (base 11)

49 (base 10) = 37 (base 14)

64 (base 10) = 40 (base 16)

100 (base 10) = 79 (base 13)

Each of these has a unique two-digit base (>10), and the numbers expressed in those new bases use only digits '0' to '9'. I make no sense of the "is unique to itself" phrase beyond that, so do not know if that is intended as a further spec. Perhaps it means that each of the digits in the higher bases must appear only once in the set of 5 conversions, but the more grammatical sense would be that it only requires five different bases, which my set satisfies (one could find other base/translations if needed.