Mathematica cannot take such integrals, even for zero-mean normal distributions. So, it is highly unlikely that they can expressed in closed form. 

Here is the image of the corresponding Mathematica notebook: 

[![enter image description here][1]][1]

An exceptional trivial case is when the two normal distributions are the same: 

[![enter image description here][2]][2]

That is, here we get $F(y)^2/2=\Phi(y)^2/2$, by the substitution $t=\Phi(x)$. 


  [1]: https://i.sstatic.net/doxBo.png
  [2]: https://i.sstatic.net/24dSd.png