We discuss the implications of a small indium content (3%) in a GaAs epilayer on the electron and nuclear spin relaxation due to enhanced quadrupolar effects induced by the strain. Using the weakly perturbative spin noise spectroscopy, we study the electron spin relaxation dynamics without explicit excitation. The observed temperature dependence indicates the presence of localized states, which have an increased interaction with the surrounding nuclear spins. Time-resolved spin noise spectroscopy is then applied to study the relaxation dynamics of the optically pumped nuclear spin system. It shows a multi-exponential decay with time components, ranging from several seconds to hundreds of seconds. Further, we provide a measurement of the local magnetic field acting between the nuclear spins and discover a strong contribution of quadrupole effects. Finally, we apply the nuclear spin diffusion model, that allows us to estimate the concentration of the localized carrier states and to determine the nuclear spin diffusion constant characteristic for this system.