We report on the magnetocaloric effect and ultrasound studies of the frustrated quasi-one-dimensional spin-1/2 compound LiCuVO4, evidencing a spin-nematic state. The magnetic Grüneisen parameter diverges at the transition to the spin-nematic phase, μ0Hc3≈40 T, showing quantum criticality accompanied by entropy accumulation. The observed high-field anomalies in the acoustic properties clearly evidence a strong involvement of the lattice in the spin dynamics. The theoretical approach, based on exchange-striction coupling with dipolar and quadrupolar contributions, suggests that the spin-dipole-strain and quadrupole-strain interactions govern the spin-nematicity in LiCuVO4.