Yes/No :number of element of order $p$ is $ p^2 +1$ Determine the  number  of element of order $p$ in $\mathbb{Z_{p^2}} \times \mathbb{Z_{p^2}}$?
My attempt :  number  of element of order $p$ in $\mathbb{Z_{p^2}} \times \mathbb{Z_{p^2}}= \frac{(p^2)^2-1}{ p^2-1}= p^2+1$
Is its  true ?

You have not shown enough details.
Let $p=|(a,b)|=\text{lcm}(a,b), a, b \in \mathbb{Z}_{p^2}.$ This is equivalent to $|a|,|b|\in \{1,p\}$ excluding the case $a=b=e,$ so we have $p^2 - 1$ such elements.
