We demonstrate the application of microwave three-wave mixing to the amino alcohol valinol, which displays a hyperfine structure in the rotational spectrum due to nuclear quadrupole coupling. The hyperfine structure complicates the typical triad of rotational states, leading to overlapping microwave three-wave mixing cycles. We identified a set of cycles accessible within the hyperfine substructure of the rotational states |JKaKc⟩ = |101⟩, |212⟩, and |202⟩ by applying the selection rules for rotational transitions and nuclear quadrupole coupling. To address an individual cycle of hyperfine transitions or subsets of cycles simultaneously, we explored different pulse schemes exploiting single-frequency or chirped microwave pulses. Each pulse scheme generated a distinct chiral signal, which shows clear enantiomer differentiation. The experimental findings agree very well with numerical simulations using an effective model for the hyperfine interaction. This study thus extends the applicability of microwave three-wave mixing to previously unexplored molecular systems containing quadrupolar nuclei.