jmc

First, we calculate that there are a total of $4\cdot 4=16$ possibilities. Now, we list all of two-digit perfect squares. $64$ and $81$ are the only ones that can be made using the spinner. Consequently, there is a $\frac{2}{16}=\boxed{\frac{1}{8}}$ probability that the number formed by the two spinners is a perfect square.