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So it's my understanding that the underlying symmetry of GR is the Diffeomorphism Group of 3+1 spacetime.
It's also my understanding that a symmetry implies a corresponding conserved quantity in a physical theory.
Then, if my understanding is correct, there should be a conserved quantity associated with the Diffeomorphism Group of 3+1 spacetime. What is it?
There seems to be some confusion regarding conservation laws vs. invariant quantities. Any (Lorentz) scalar, such as density, pressure, temperature, or charge, will be invariant as reference frames change. So too will any vector or higher other tensorial quantity. That is, even though different observers might assign different numbers to the components of these objects, they all agree that the objects are simply the same thing represented in different bases.
A conservation law says that something is neither created nor destroyed but only moves around as the system evolves. Since time is just a part of spacetime, in relativity this amounts to the vanishing of a covariant divergence: $\nabla_\mu Q^\mu = 0$ for some $Q$ (which may have other components). All the conservation laws you know and love are conserved in this local sense in GR: particle number density in a fluid ($Q^\mu = nu^\mu$ with $u^\mu$ the components of the fluid's 4-velocity, assuming there is no annihilation/production), rest mass density ($Q^\mu = \rho u^\mu$, again under that assumption), energy and momentum ($Q^\mu = T^{\mu\nu}$, with no further assumptions needed), etc.
(This answer was written before this question was drastically rewritten, when the basic question was "Is it possible to define a conserved quantity in GR?".)
Sure, there are derivative forms of conservation laws that hold in general relativity. For example, there's the conservation of the stress-energy tensor,
$${T^{\mu\nu}}_{;\nu} =0\ .$$
