The cdf of a two-dimensional Farlie-Gumbel-Morgenstern distribution is given by
with $|\lambda|<1$ and $\alpha_i,\beta_i>0$ for $i=1,2$. This implies that, if $(X,Y)\sim F$, then $...
Just consider the conditional distribution of $Y|Z$. This can be characterized as
$P(Y = z|Z = z) = 0.5$ and $P(Y = -z|Z = z) = 0.5$. This is no normal distribution and therefore the distribution of $(Y, Z)$ is not bivariate normal (because for a multivariate normal distribution the conditional distributions are also normal).