Does the quantum nature of light arise from its interaction with matter? I have a desire to reconcile the results of the photoelectric effect with the Maxwellian picture of electromagnetic radiation. I wish to explore, the possibility that the quantum nature of the photon arises from its interaction with matter - specifically perhaps, arising from light's interaction with the discrete electron orbitals of matter; and that light otherwise, is indeed an electromagnetic wave (rather than a particle) at a fundamental level, as described by Maxwell.
Could this idea reconcile the quantum results of the photoelectric effect experiment with the classical view of light as a continuous electromagnetic wave as described by Maxwell and not a corpuscle as described by Newton and concluded by Einstein from these results?
Perhaps we also need to consider the source of the light.  Is it for example produced by matter as in a tungsten filament or radiation from a glowing object?  In those cases, of course we might expect light will be emitted as quanta of energy /photons/ particles. However, if the EMR is generated by fluctuating the voltage between two electric plates or fluctuating a magnetic field in space, then surely this light should indeed be a pure wave!  Would we still succeed in detecting photon nature of light if we generated light without matter and detected light without matter, that is did not involve any interaction with matter?
 A: If we want a correct treatment of electromagnetic field then we are going to have to include a correct treatment of effects such as the Lamb shift, the gyromagnetic ratio of the electron, and things like that. From your question I am guessing you are not very familiar with those. They are effects that are very accurately treated by the quantum field theory called quantum electrodynamics, in which light is neither wave nor particle but quantum field.
So to answer your question, it is true that many of the physical effects which tend to be mentioned in the context of photons concern the way light interacts with matter, and if the matter has discrete energy levels then you might feel that the discreteness lies only there. But you should also consider for example the Compton effect, where light scatters off electrons in free space. Here there is no discreteness in the states of the electron, but explaining the results in terms of scattering waves is hard to do, whereas the results are readily explained on a particle model. The fully accurate model is the quantum field theory one, as I already said, in which light is neither particle nor wave. And to understand the Lamb shift, the Casimir effect, the electro-weak theory, and all the rest of moden particle physics it is almost certainly hopeless to try a classical-field type of model for electromagnetism.
A: Indeed, one way to "visualize the unvisualizable" is to think of light in flight as consisting of photons, which when called upon to interact with charged matter, begin exhibiting wavelike characteristics in one set of circumstances (double-slit diffraction, for example) and yet retain their photon-like characteristics in another set of circumstances (photoelectric effects).
And as pointed out by Anna V, a closer examination of just the double-slit, single-photon diffraction experiment reveals the same object (the photon/wave) behaving like a photon at one point in time and like a wave at another point in time (see her linked reference below).
I do not know if this picture of the unknowable can be made fully consistent with Maxwellian electrodynamics, but it is the only way that I can wrap my brain around particle-wave duality.
A: 
Would we still succeed in detecting photon nature of light if we
generated light without matter and detected light without matter, that
is did not involve any interaction with matter?

$c$ is the rate at which "elsewhere and in the future" turns into "here and now" and then into "elsewhere and in the past". Outgoing light (and other massless things) go irretrievably from "here and now" to elsewhere in the future, while incoming light comes here exclusively from elsewhere in the past.
If we don't use a massive (that is to say: $v<c$) detector at some point in the process, the interaction of our detector and our photon is always somewhere else and in our observer's future. It is never observed.
Likewise, if we don't use a massive source at some point in the process, our source is always somewhere else and in our past, so our experiment has to start infinitely far away and infinitely long ago.
We can push the matter behind some layers of abstraction like "voltage" and "magnetic moment" and make-believe that its not made of huge numbers of massive particles interacting with one another and the space around them, but that's a logic error.
We can generate microwaves with a magnetron and model the system with great predictive efficacy by pretending that the magnetron is a single magnetic dipole, not countless atoms each with a magnetic moment - but we know that the latter is a better description of reality.
We can measure an interference pattern and pretend that what we have measured is the wave, not the wave's effect on a sensor, with great descriptive efficacy. But we know that the latter is what really happened.
A: There were two revolutions in physics in the early 20th C. The first led by Einstein and others revolutionised our understanding of space and time. The second, the quantum revolution, transformed our understanding of how things are - that is ontology. Before this revolution, things were primarily understood as either particles in space or waves in a medium. After, they were understood in terms of quantum fields supervening on spacetime and whose excitations were the quanta.

Does the quantum nature of light arise from its interactions with matter?

Since light can exist in a vacuum (think of light travelling from the sun) it's difficult to see how the quantum nature of light arises due to its interactions with matter.
A: The premise of the question is incorrect, as there is no contradiction between the Maxwellian theory and the quantization of light.
Quantized EM field is consistent with Maxwell equations
Quantization of electromagnetic field imposes non-commutativity between different parts of the field. Mathematically it is achieved by transforming the amplitudes of the field, when the field is expanded in terms of its eigenmodes. These eigenmodes are the solutions of Maxwell equations. In other words, quantized EM field automatically satisfies Maxwell equations.
Coherent states
Another link is provided by the Ehrenfest theorem, which for the quantized EM fields takes the form of the Maxwell equations. In particular, the quantized EM field is close to the wave-like properties of the classical EM field when it is in a coherent state.
Wave-particle duality
Finally, it is incorrect to think of a photon as a point-like particle, similar to an electron - rather it is an excitation quantum of the EM field. Quantization does not make photons into particles or electrons into waves - rather it means that both exhibit wave-like and particle-like properties - this what we call wave-particle duality see here and here)
