The discovery of superconducting transitions in pressurized ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$ and ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_{10}$ has highlighted the pivotal role of density wave (DW) orders in nickelate superconductors. To gain a comprehensive understanding of the superconducting state, it is essential to elucidate the nature of the DW order. In this study, we utilized ${}^{139}\mathrm{La}$ nuclear quadrupole resonance (NQR) to investigate the charge-density wave (CDW) and spin-density wave (SDW) orders in both single-crystal and polycrystalline ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_{10}$. Near $T_{\mathrm{DW}}\approx 133\mathrm{K}$, an abrupt change in both the linewidth and frequency of the La(2) site in the single-crystal sample provides compelling evidence for a first-order-like phase transition. The pronounced broadening of the NQR lines indicates the incommensurate nature of the DW order. Furthermore, the spin-lattice relaxation rate divided by temperature 1/$T_{1}T$ exhibits a strong enhancement at $T_{\mathrm{DW}}$, indicating the strong spin fluctuations above the first-order DW transition. These observations suggest an intricate interplay between incommensurate CDW and SDW orders. Our findings offer critical insights into the microscopic mechanisms of the DW state in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_{10}$ and establish an essential framework for exploring the interplay between DW and superconducting phases in nickelate superconductors.

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• Charge density waves
• Spin density waves
• High-temperature superconductors
• Nuclear quadrupole resonance