The Gaia Alerts system is providing alerts on a variety of objects displaying a significant photometric change detected by the Gaia satellite from one passage to the next one over the same region of the sky. Among the ∼23,000 alerts published until the end of 2022, around 13% concern AGN or quasar candidates. At the same time, a different method to detect variations specifically in galactic nuclei was tested on Gaia data during a one year period only, leading to another set of candidates. We have embarked on a spectroscopic ground-based follow-up of some of those, to establish their true nature, and reveal the various phenomena leading to a change in magnitude of those AGN. The present paper deals with radio-quiet objects, while the radio-loud ones will be described in a companion paper. We confirm, on one hand, the AGN/quasar nature of 64 new candidates alerted by Gaia, and, on the other hand, obtained second-epoch spectra of over 200 known AGN also alerted for large photometric variations. The observed phenomena show a large variety described in this paper: from Flares to Tidal Disruption Events (TDEs) and a large number of Changing Look Quasars (CLQs, 56 secure ones, plus 23 probable ones), not forgetting some rarer events like SNe, microlensing events or Extreme Coronal Line Emitters. This sample shows that variability is an excellent tool to detect new quasars, especially radio-quiet ones that otherwise would be undetected, and that a significant fraction of variable AGN/quasars, around 10%, presents the CLQ phenomenon. Some of the new CLQs are followed-up to monitor further changes and measure time scales.