0² ends in a 0
1² ends in a 1 
2² ends in a 4
3² ends in a 9
4² ends in a 6
5² ends in a 5
6² ends in a 6
7² ends in a 9
8² ends in a 4
9² ends in a 1

So the only cases on the above list where the square of a 
digit ends in that digit are these 4

0² ends in a 0
1² ends in a 1
5² ends in a 5
6² ends in a 6

The numbers 0 to 90 that end in 0 are 0,10,20,...90, that's 10.
The numbers 0 to 90 that end in 1 are 1,11,21,...81, that's 9.
The numbers 0 to 90 that end in 5 are 5,15,25,...85, that's 9.
The numbers 0 to 90 that end in 6 are 6,16,26,...86, that's 9.

10+9+9+9 = 37

Edwin