In general relativity, it doesn't really make sense to talk about a hypersurface of constant time as if that had some intrinsic physical meaning. Coordinates such as a time coordinate are arbitrary in GR. So to state your question correctly, what you really want to talk about is a spacelike surface or Cauchy surface. Let's call this surface S.
The answer to your question is no, for physically reasonable matter fields. The field equations imply that the stress-energy T has zero divergence, which is a local statement of conservation of energy-momentum. We don't have a global Gauss's law in a curved spacetime, but local conservation is enough to rule out your scenario. For example, suppose that a hydrogen atom pops into existence at some point in spacetime, as in the old steady-state cosmological models. For a sufficiently small neighborhood of this point, curvature is negligible, and we can adopt Minkowski coordinates in which the atom is at rest. In these coordinates, $\partial_\kappa T^{\kappa t}=\partial T^{tt}/\partial t\ne 0$.
The idea here is that although it is possible to trade gravitational energy (which is not counted in the stress-energy) for the energy of matter fields, we must always do so in such a way that to a local, free-falling observer, energy appears to be conserved. This is the equivalence principle.
What I gave above is only an argument that rules out one specific example, in which a hydrogen atom spontaneously pops into existence. The only fact about this matter field that was used in the argument was that it was possible to define a local Minkowski frame in which the matter field was at rest. Referring to an answer to a very similar question, I think this is equivalent to assuming the dominant energy condition. The DEC ensures that the flow of energy is subluminal, so that we can define such a frame.
There are independent physical reasons why such a scenario is problematic. We expect the matter fields to obey some wave equation, but if the wave and all its derivatives are zero on a Cauchy surface, then it would seem to violate causality if the wave were later to be nonzero. Also, it will be impossible to avoid a violation of Lorentz invariance, since there is no preferred frame of reference in which a newly created particle should be at rest.
AVS says in a comment:
Electron–positron pair could certainly annihilate into a pair of gravitons. This process would have a corresponding solution of (classical) Einstein–Dirac system.
This would seem to violate Lemma 4.3.1 in Hawking and Ellis, which they say on p. 94 can be interpreted to give this:
...if the energy-momentum tensor vanishes on [a Cauchy surface], then it also vanishes on the future Cauchy development [of that set]
(Cf. Wald, p. 219.) It would be interesting to understand how the counterexample that you sketch evades this. Maybe you could develop this example a little more clearly as a separate question.