Suppose I have $X,Y$ bivariate normal with correlation coefficient $\rho \in (0,1)$ . Then , what is the correlation between $X^2 $ and $Y^2$ ?


I am aware of the fact that the square of the normal follows a chi square distribution . So , I can find out $Var(X^2)$ and $Var(Y^2)$ . However , I am unable to calculate the covariance between $X^2$ and $Y^2$ .



Correlation between square of normal random variables
You can do what Nate Eldredge suggested. Otherwise, you can use the moment generating function $M$ of the bivariate normal distribution $N(\mu\_1,\mu\_2,\sigma\_1^2,\sigma\_2^2,\rho)$ of $(X,Y)$ given by 
$$M(t\_1,t\_2)=\exp(\boldsymbol\mu'\boldsymbol t+\tfrac12\,\boldsymbol t'\Sigma\boldsymbol t)
$$
for $\boldsymbol t=[t\_1,t\_2]'$, $\boldsymbol\mu=[\mu\_1,\mu\_2]'$, 
$\Sigma=\begin{bmatrix}\sigma\_1^2&\rho\sigma\_1\sigma\_2\\\rho\sigma\_1\sigma\_2&\sigma\_2^2 \end{bmatrix}$, where $'$ denotes the transpose of a matrix. 
Then 
$$EX^2Y^2=\frac{\partial^4}{\partial^2t\_1\,\partial^2t\_2}\,M(t\_1,t\_2)\Big|\_{t\_1=t\_2=0}=
4 \mu \_1 \mu \_2 \rho \sigma \_1 \sigma \_2+\left(\mu \_1^2+\sigma \_1^2\right) \left(\mu
 \_2^2+\sigma \_2^2\right)+2 \rho ^2 \sigma \_1^2 \sigma \_2^2,
$$ 
whence 
$$Cov(X^2,Y^2)=EX^2Y^2-EX^2\,EY^2=4 \mu \_1 \mu \_2 \rho \sigma \_1 \sigma \_2+2 \rho ^2 \sigma \_1^2 \sigma \_2^2. 
$$
Special cases of this formula are 
$$Var(X^2)=Cov(X^2,X^2)=4 \mu \_1^2 \sigma \_1^2+2 \sigma \_1^4 
$$
and 
$$Var(Y^2)=Cov(Y^2,Y^2)=4 \mu \_2^2 \sigma \_2^2+2 \sigma \_2^4. 
$$
So, the correlation between $X^2$ and $Y^2$ is 
$$\frac{Cov(X^2,Y^2)}{\sqrt{Var(X^2)Var(Y^2)}}
=\frac{4 \mu \_1 \mu \_2 \rho \sigma \_1 \sigma \_2+2 \rho ^2 \sigma \_1^2 \sigma
 \_2^2}{\sqrt{\left(4 \mu \_1^2 \sigma \_1^2+2 \sigma \_1^4\right) \left(4 \mu \_2^2 \sigma
 \_2^2+2 \sigma \_2^4\right)}}
=\frac{2 \mu \_1 \mu \_2 \rho \sigma \_1 \sigma \_2+ \rho ^2 \sigma \_1^2 \sigma
 \_2^2}{\sqrt{\left(2 \mu \_1^2 \sigma \_1^2+ \sigma \_1^4\right) \left(2 \mu \_2^2 \sigma
 \_2^2+ \sigma \_2^4\right)}}.
$$
In particular, the correlation between $X^2$ and $Y^2$ is $\rho^2$ when $\mu\_1=\mu\_2=0$. 


