There is some confusion here:
Matter and energy are related; one can convert into the other. What is it called when this happens?
Matter in the sense of "mass" is related to energy through the famous E=m*c**2 of Special Relativity. That is the only framework where conversion of matter to energy has a meaning in physics.
For example, solids melt/liquefy into liquid, and liquid vaporizes into gas. Gas condenses into liquid, and liquid solidifies into solid.
is out of classical physics, thermodynamics, and it is talking of phase transitions of matter. There is no change of matter to energy or energy to matter in your example. There is only a change in the form of energy, from kinetic to potential, binding the individual atoms and molecules dependent on the temperature . So solids turn into liquids and liquids to gas depending on temperature. "turn" is a better turn of language :) to "transition" which is the physics term.
Then again you go into a phrasing that is appropriate for special relativity:
So matter [verb] into energy, and energy [verb] into matter.
Here you can use "converts" if you are discussing fusion and fission reactions, mass converts to energy, also "becomes" and "turns" are not wrong turns of phrase. If an energetic photon creates an electron positron pair in the field of an atom or nucleus, "the energy of the photon partially converts to matter" is fine ( a lot of the incoming energy goes to kinetic energy, momentum balances, and a scattered photon of less energy).
What are the correct verbs to use here? I am thinking that they are "energize" and "materialize", but I'm not sure, especially about "energize".
"energize" is wrong, "materialize" is too science fiction.