We can expand aabb as

=> 1000a + 100a + 10b + b
= 1100a + 11b = 11(100a + b)

for 11(100a +b) to be a perfect square 100a + b should be a multiple of 11 where a and b both are integers and single digit numbers.

lets say 100a + b = 11y, here y should also be a perfect square (such that 11*11*y is a perfect square), i.e. 1, 4, 9, 16, 25, 36

If you try to solve this, where a and b are single digit integers and y is also a perfect square, you can see one possible values of a and b are 7 and 4, where y = 64.

Thus the number should be 7744, it is a perfect square of 88.