We present constraints on local primordial non-Gaussianity (PNG), parametrized through f(NL)(loc), using the Sloan Digital Sky Survey IV extended Baryon Oscillation Spectroscopic Survey Data Release 14 quasar sample. We measure and analyze the anisotropic clustering of the quasars in Fourier space, testing for the scale-dependent bias introduced by primordial non-Gaussianity on large scales. We derive and employ a power spectrum estimator using optimal weights that account for the redshift evolution of the PNG signal. We find constraints of -51 < f(NL)(loc), < 21 at 95% confidence level. These are among the tightest constraints from Large Scale Structure (LSS) data. Our redshift weighting improves the error bar by 15% in comparison to the unweighted case. If quasars have lower response to PNG, the constraint degrades to -81 < f(NL)(loc) < 26, with a 40% improvement over the standard approach. We forecast that the full eBOSS dataset could reach sigma( fNLloc) similar or equal to 5-8 using optimal methods and full range of scales.