The indirect NMR detection of quadrupolar nuclei in solids under magic-angle spinning (MAS) is possible with the through-space HMQC (heteronuclear multiple-quantum coherence) scheme incorporating the TRAPDOR (transfer of population in double-resonance) dipolar recoupling. This sequence, called T-HMQC, exhibits limited t1-noise. In this contribution, with the help of numerical simulations of spin dynamics, we show that most of the time, the fastest coherence transfer in the T-HMQC scheme is achieved when TRAPDOR recoupling employs the highest radiofrequency (rf) field compatible with the probe specifications. We also demonstrate how the indirect detection of the triple-quantum (3Q) coherences of spin-3/2 quadrupolar nuclei in solids improves the spectral resolution for these isotopes. The sequence is then called T-HMQC3. We demonstrate the gain in resolution provided by this sequence for the indirect proton detection of 35Cl nuclei in l-histidine∙HCl and l-cysteine∙HCl, as well as that of 23Na isotope in NaH2PO4. These experiments indicate that the gain in resolution depends on the relative values of the chemical and quadrupolar-induced shifts (QIS) for the different spin-3/2 species. In the case of NaH2PO4, we show that the transfer efficiency of the T-HMQC3 sequence employing an rf-field of 80 kHz with a MAS frequency of 62.5 kHz reaches 75% of that of the t1-noise eliminated (TONE) dipolar-mediated HMQC (D-HMQC) scheme.