What is the fate of a photonic quantum that hits a black wall?

There is no conservation of matter , as photons belong to the special relativity and quantum mechanic frame. Only energy and momentum and angular momentum conservation laws are applicable.

A black wall is black because it absorbs the photons impinging on it. Depending on the frequency of the photon its energy is given up to displacing an electron in an atomic or molecular or lattice level to a different energy level and the relaxation to lower levels releases photons that are further absorbed. Momentum is conserved by the wall mass.

Note that black is a term related to the frequency of light that the photon belongs to. No matter is black for all frequencies, that is why we can have infrared photography.

The black wall would be composed of molecule that absorbs light at the wavelength of the photon. So the photon (quantum state and all) gets absorbed by one such molecule and the energy and momentum gets transferred to the molecule. Now there are a plethora of ways in which the energy and momentum can be dissipated. In general the energy is converted into heat that is dissipated into the wall or perhaps re-radiate back into the atmosphere. The momentum would be adding to the momentum of the wall. However, the amount of energy and momentum that the wall receives from one photon is so tiny that it makes an insignificant difference.