In the past 5 yr, six X-ray quasi-periodic eruption (QPE) sources have been discovered in the nuclei of nearby galaxies. Their origin remains an open question. We present Multi Unit Spectroscopic Explorer integral field spectroscopy of five QPE host galaxies to characterize their properties. We find that 3/5 galaxies host extended emission-line regions (EELRs) up to 10 kpc in size. The EELRs are photoionized by a nonstellar continuum, but the current nuclear luminosity is insufficient to power the observed emission lines. The EELRs are decoupled from the stars both kinematically and in projected sky position, and the low velocities and velocity dispersions (<100 km s(-1) and less than or similar to 75 km s(-1), respectively) are inconsistent with being driven by active galactic nuclei (AGNs) or shocks. The origin of the EELRs is likely a previous phase of nuclear activity. QPE host galaxies share several similarities with tidal disruption event (TDE) hosts, including an overrepresentation of galaxies with strong Balmer absorption and little ongoing star formation, as well as a preference for a short-lived (the typical EELR lifetime is similar to 15,000 yr), gas-rich phase where the nucleus has recently faded significantly. This suggests that QPEs and TDEs may share a common formation channel, disfavoring AGN accretion disk instabilities as the origin of QPEs. If QPEs are related to extreme mass ratio inspiral systems (EMRIs), e.g., stellar-mass objects on bound orbits about massive black holes, the high incidence of EELRs and recently faded nuclei could be used to localize the hosts of EMRIs discovered by low-frequency gravitational-wave observatories.