Conservation of energy and information are partly related to each other, but in a weak and indirect way where one does not imply the other. Conservation of energy comes from time symmetry in physics ignoring the effects of general relativity (where it is not conserved in arbitrary spacetimes because there is no time symmetry in all of them. An example is the cosmological standard model). Thus it is conserved in quantum field theory (for the most part, T symmetry is possibly weakly not true in the weak interactions).
Information conservation is also mostly conserved, but for a different reason, and means something different. First the mostly: for black holes it is still not clear how it would be conserved, since as particles fall in everything about them seems to disappear - but it is still a a paradox and controversial, and really unresolved. What is that information that is conserved? It is the probabilities of all the quantum states that make up all the states of a quantum system. When a particle interacts with another the states may change, but they do so in such a way that is determined by the system Hamiltonian, and requires that the system evolution be unitary. That conserves the total probability of the state of the system, even as the state evolves. That, and variation and details of that are the qubits that make us quantum information. It could include information on the energy of the system, and if time is symmetric that will be one of the quantum information bits, the qubits, that will be conserved.
So, one could argue that the quantum state information, which may include energy, is conserved. That is all of what one can really conclude. Sussking may mean this, or may be trying to make deeper comments, about that. Information conservation is simply conservation of physical state probabilities when they all get added up. See some of the related questions on the right.
Still, quantum information work deals more with how to use the physical state information, the qubits, to compute much faster than in classical computing. But the Black Hole issue I mentioned above, the Black Hole paradox has introduced physics discussions that (such as what is true then, General Relativity or Quantum Theory, the paradox seems to say both cannot be true). It is also a good way to try to understand what quantum information conservation really means. See https://en.m.wikipedia.org/wiki/Black_hole_information_paradox
There are many references that talk about quantum information. See as one
So, yes, they are related somewhat, but really you can derive energy conservation from information conservation. The former relies on time symmetry, the latter on the unitarity of quantum theory. They are different things.