Squares from 1 to 10 is easy.
For squares from 11 to 19 you can adapt the following method-
Say, 132
Take the unit digit and and add it to the number itself.
13 + 3 = 16
This will become the tens and hundreds digit of the squared number.
Now, square the unit digit,
32=9
This will become the unit digit of the squared number.
So, the square of 13 is 169.
Similarly you can check for other numbers.
This is valid from 11 to 19 only.