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Quantum Einstein Gravity (QEG), nonperturbatively renormalized by means of a certain asymptotically safe renormalization group (RG) trajectory, is explored by solving its scale dependent effective field equations and embedding the family of emerging 4-dimensional spacetimes into a single 5-dimensional manifold, which thus encodes the complete information about all scales. By construction the latter manifold is furnished with a natural foliation. Heuristically, its leaves are interpreted as physical spacetime observed on different scales of the experimental resolution. Generalizing earlier work on the embedding of $d$-dimensional Euclidean QEG spacetimes in $(d + 1)$-dimensional flat or Ricci flat manifolds, we admit Lorentzian signature in this paper and we consider embeddings in arbitrary $(d + 1)$-dimensional Einstein spaces. Special attention is paid to the sector of maximally symmetric metrics, and the fundamental definition of QEG in $d = 4$ that employs the cross-over trajectory connecting the non-Gaussian to the Gaussian RG fixed point. Concerning the embedding of the resulting family of 4D de Sitter solutions with a running Hubble parameter, we find that there are only two possible 5D spacetimes, namely the anti-de Sitter manifold AdS$_5$ and the de Sitter manifold dS$_5$. To arrive at this result essential use is made of the monotone scale dependence of the running cosmological constant featured by the gravitational effective average action. We show that if the scale invariance of the QEG fixed points extends to full conformal invariance, the 5D picture of the resulting geometric and field theoretic structure displays a novel kind of "AdS/CFT correspondence". While strongly reminiscent of the usual string theory-based AdS/CFT correspondence, also clear differences are found.