Entanglement is a key property of quantum systems. In this Letter the first measurements of quantum entanglement between spins in pairs of $Z$ bosons are reported, using proton-proton collision data from the Large Hadron Collider (LHC) at center-of-mass energies of 13 and 13.6 TeV, recorded with the ATLAS detector. Measurements of angular observables sensitive to $Z{Z}^{*}$ spin-density-matrix elements in the $H\to Z{Z}^{*}\to {\ell }^{+}{\ell }^{-}{\ell }^{+}{\ell }^{-}$ process yield coefficients $C_{2,1,2,-1}=-0.71\pm 0.45$ and $C_{2,2,2,-2}=0.08\pm 0.44$, consistent with their Standard Model predictions. A complementary hypothesis test using the full angular distribution, and relying on several Standard Model assumptions in the decays, provides substantially higher sensitivity to quantum correlations and disfavors the separable-state hypothesis at a significance of 4.7 standard deviations (expected $4.9\sigma$) relative to the entangled Standard Model hypothesis. These results provide strong evidence of quantum entanglement between massive bosons (spin qutrits) at the electroweak scale.