  Motivated by the largely unexplored domain of multi-polar ordered spin states
in the Kitaev-Heisenberg (KH) systems we investigate the ground state dynamics
of the spin-$\frac{1}{2}$ KH model, focusing on quadrupolar (QP) order in 2-leg
ladder and two-dimensional honeycomb lattice geometries. Employing exact
diagonalization and density-matrix renormalization group methods, we analyze
the QP order parameter and correlation functions. Our findings reveal a robust
QP order across a wide range of the phase diagram, influenced by the interplay
between Heisenberg and Kitaev interactions. Notably, we observe an enhancement
of QP order near Kitaev quantum spin liquid (QSL) phases, despite the absence
of long-range spin-spin correlations. This highlights a complex relationship
between QP order and QSLs, offering new insights into quantum magnetism in
low-dimensional systems. Our findings provide a rational explanation for the
observed nonlinear magnetic susceptibility in $\alpha$-RuCl$_3$.
