Short answer: No. Momentum-energy conservation arises by dint of Noether's theorem, which says that if a system's Lagrangian is invariant with respect to a continuous transformation, there is one conserved quantity, called the "Noether Charge" for each such transformation (technically: for each linearly independent tangent vector in the Lie algebra of the group of such transformation).
Momentum-energy is the Noether charge corresponding to the continuous "symmetry" of spacetime translation. If the physics of the physical system you're dealing with doesn't change if you continuously shift the origin of your spacetime co-ordinates, then we get momentum-energy conservation. (there is one Noether charge for each part of this shift: $p_x,\,p_y,\,p_z$ for shifts in the $X,\,Y,\,Z$ direction and energy for time-translation).
General solutions of the Einstein field equations do not have these symmetries. No global time or space co-ordinates are generally definable. Spacetime over here has, in general, a different curvature from spacetime over there, so the physics does care where we put our origin.