aabb 

is 1000a+100a+10b+b

and can be re written as 1100a+11b or

11(100a+b)

So for the no. to be a perfect square 100a+b should be  a no like 11n

where n is a perfect square.

The maximum possibe no with 100a+b is 909.

But as we know last digit of any square no. should end in either of 

0,1,4,9,6,5

and the Nos. are ( a cant be 0 , if so, aabb will no longerbe 4 digited )   

                         100,200....900            

                         101,201....901

                         104.......... 904 

                         109.....      909 

We are lucky here that We just need to add first and last digit of the numbers  for checking div by 11.Now we are left with only 

704 , 605, 506, 209  of which only 704 = 11*64 gives the value of n as a perfect square .

Making your  a as 7, and b as 4  and the no . is 7744 .

Leaving me with option 5. 1 no.