Or maybe if I have two cells, with the starting integer in A1 and the
ending integer in A2 the funciton would look as follows:
SUMFUCNC((A1:A2)^2). This would also return 1+4+9+...+100.

Let's see A1 for first integer, and B1 as last integer, use this array
formula:
=SUM(ROW(INDIRECT("A"&INDIRECT("A1")&":B"&INDIRECT ("B1")))^2)

Regards,
Daniel