We present spatially resolved, SSP-equivalent ages, stellar metallicities, and abundance ratios for 456 massive (10.3 ≲ log (M*/M⊙) ≲ 11.8) quiescent galaxies at 0.6 ≲ z ≲ 1.0 from the LEGA-C survey, derived using full-spectrum models. Typically, we find flat age and [Mg/Fe] gradients, and negative [Fe/H] gradients, implying iron-rich cores. We also estimate intrinsic [Fe/H] gradients via forward-modeling. We examine the observed gradients in three age bins. Younger quiescent galaxies typically have negative [Fe/H] gradients and positive age gradients, possibly indicating a recent central starburst. Additionally, this finding suggests that photometrically-measured flat colour gradients in young quiescent galaxies are the result of the positive age and negative metallicity gradients cancelling each other. For older quiescent galaxies, the age gradients become flat and [Fe/H] gradients weaken, though remain negative. Thus, negative colour gradients at older ages are likely driven by metallicity gradients. The diminishing age gradient may result from the starburst fading. Furthermore, the persistence of the [Fe/H] gradients may suggest that the outskirts are simultaneously built up by mergers with lower-metallicity satellites. On the other hand, the gradients could be inherited from the star-forming phase, in which case mergers may not be needed to explain our findings. This work illustrates the need for resolved spectroscopy, instead of just photometry, to measure stellar population gradients. Extending these measurements to higher redshift is imperative for understanding how stellar populations in quiescent galaxies are assembled over cosmic time.