Motivated by the largely unexplored domain of multipolar ordered spin states in the Kitaev-Heisenberg (KH) systems we investigate the ground state dynamics of the spin-12 KH model, focusing on quadrupolar (QP) order in two-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 different insights into quantum magnetism in low-dimensional systems. Our findings provide a rational explanation for the observed nonlinear magnetic susceptibility in α-RuCl3.