Does the fact that energy is not conserved in cosmology opens the possibility of new matter/atoms being created in the universe?
Creating new baryonic matter is something that was considered seriously in the context of the steady state model, which was advocated by Hoyle, Bondi, and Gold up until about the mid-1960's, when evidence for a hot big bang killed the steady state model off. One of the problems with steady state models was that they had to violate Lorentz invariance, since there had to be some preferred state of motion for the newly created hydrogen atoms. (They also violate the charge conjugation and time reversal symmetries.)
You're right that mass-energy is not globally conserved in general relativity, but it is exactly locally conserved in the sense that the divergence $\nabla_a T^a{}_b$ of the stress-energy tensor is zero.
Our universe is currently pretty well approximated by the de Sitter spacetime, and is expected to remain so forever, according to current theories. For simplicity, let's suppose for the moment that the geometry of the universe has to be exactly de Sitter. This was what Hoyle et al. wanted, because they wanted all eras of the universe to look the same, and the de Sitter universe is the only cosmological model that has this symmetry. This seems pretty similar to the physical motivation for your question, which was whether it would be possible for the universe to avoid a fate in which there was basically nothing there but dark energy.
For the de Sitter spacetime the divergence of the stress-energy tensor has a timelike component equal to
$$\frac{\dot{a}}{a}(\rho+P),$$
where $a$ is the scale factor describing cosmological expansion, $\rho$ is the mass-energy density, and $P$ is the pressure. (This is all in units where $c=1$.) For this reason, we must have
$$\rho+P=0$$
everywhere. Dark energy satisfies this condition, but baryonic matter doesn't. Therefore it is not possible for new baryonic matter to be created in cosmological expansion, in the de Sitter spacetime.
If you're a wily theoretician like Hoyle and you look for a way to escape this constraint, there is a way out, which is to posit the existence of a field with $\rho=0$ and $P<0$. Hoyle called this the C field. Then if you add the contributions to the stress-energy from the C field and baryonic matter, you can end up with $\rho+P=0$.
Although dark energy is currently the dominant form of mass-energy in the universe, realistic cosmological models do incorporate other matter fields, including baryonic matter. These models therefore do not have exactly the geometry of de Sitter space. That complicates things compared to the argument given above, but the conclusion is still the same. According to these models, you can't have the production of new baryonic matter without violating local conservation of mass-energy, which is baked in to the structure of general relativity.
If you try to construct a model that produces new baryonic matter without violating local conservation of mass-energy, then as far as I know you are uniquely led to something like Hoyle's "C-field" theory, and then you have all the problems of that theory, including violation of Lorentz invariance and incompatibility with observations such as the cosmic microwave background. For more information on (failed) attempts to reconcile such theories with modern knowledge, see this web page by Ned Wright. I also have a mathematical discussion of the steady-state model in section 8.4 of my own GR book.
What raised my question is the idea of Eternal Inflation. New matter is being created in the newly created bubble universes. Of course this is highly speculative but as a layman I wonder if it’s possible since energy is not conserved.
Energy is locally conserved. It's just not globally conserved. John Rennie's answer explains why inflation isn't an exception to this.
