Reverse the number and it's square!

If $\overline {AB} \geq 15$ , then first digit of $\overline {AB} ^{2}$ will not be $A$ .

$\implies$ $\overline {AB} \leq 14$

Also if $\overline {AB} \leq 9$ , then $\overline {AB} ^{2}$ will have 2 digits.

$\implies$ $\overline {AB} \geq 10$

Hence $10 \leq\overline {AB} \leq 14$ .

Thus we have only few values to check.Putting $13 = \overline{AB}$ , we get our desired result.