Bivariate natural exponential families with quadratic diagonal of the variance function

We characterize bivariate natural exponential families having the diagonal of the variance function of the form \[ \textrm{diag} V(m_1,m_2)=\left(Am_1^2+am_1+bm_2+e,Am_2^2+cm_1+dm_2+f\right), \] with $A<0$ and $a,\ldots,f\in\mathbb{R}$. The solution of the problem relies on finding the conditions under which a specific parametric family of functions consists of Laplace transforms of some probability measures.