  This paper presents a global study of the class $\bf{Q}{\widehat{ES}}$ of all
real quadratic polynomial differential systems possessing exactly one elemental
infinite singular point and one triple infinite singular point, which is either
an infinite nilpotent elliptic-saddle or a nilpotent saddle. This class can be
divided into three different families, namely, $\bf{Q}{{\widehat{ES}(A)}}$ of
phase portraits possessing three real finite singular points,
$\bf{Q}{{\widehat{ES}(B)}}$ of phase portraits possessing one real and two
complex finite singular points, and $\bf{Q}{{\widehat{ES}(C)}}$ of phase
portraits possessing one real triple finite singular point. Here we provide the
complete study of the geometry of these three families. Modulo the action of
the affine group and time homotheties, families $\bf{Q}{{\widehat{ES}(A)}}$ and
$\bf{Q}{{\widehat{ES}(B)}}$ are three-dimensional and family
$\bf{Q}{{\widehat{ES}(C)}}$ is two-dimensional. We study the respective
bifurcation diagrams of their closures with respect to specific normal forms,
in subsets of real Euclidean spaces. The bifurcation diagram of family
$\bf{Q}{{\widehat{ES}(A)}}$ (respectively, $\bf{Q}{{\widehat{ES}(B)}}$ and
$\bf{Q}{{\widehat{ES}(C)}}$) yields 1274 (respectively, 89 and 14) subsets with
91 (respectively, 27 and 12) topologically distinct phase portraits for systems
in the closure $\overline{\bf{Q}{{\widehat{ES}(A)}}}$ (respectively,
$\overline{\bf{Q}{{\widehat{ES}(B)}}}$ and
$\overline{\bf{Q}{{\widehat{ES}(C)}}}$) within the representatives of
$\bf{Q}{{\widehat{ES}(A)}}$ (respectively, $\bf{Q}{{\widehat{ES}(B)}}$ and
$\bf{Q}{{\widehat{ES}(C)}}$) given by a specific normal form.
