If a photon can be said to change speed at a boundary between two optical materials (e.g., glass and air), then you might ask how quickly it changes speed at that boundary. However, no physical boundary is infinitesimally thin, so at least on a sub-femtosecond timescale, the effective speed of a photon changes somewhat gradually while crossing a physical boundary. But as the other answers have indicated, the "speed of a photon" is not precisely definable in a physically useful way because a photon is not really a point particle; it's a spread-out wavepacket. If the speed can't be defined precisely, the acceleration can's be defined precisely.
You didn't ask about acceleration of a point charge like an electron. We can come a lot closer to defining velocity and acceleration for an electron, but ultimately run into the same limitations due to the fact that an electron, too, is really a wavepacket. However in the case of an electron there is a further limitation: an accelerating charge radiates EM radiation at a rate proportional to the square of its acceleration See Larmor Radiation. So, the quicker you try to change its speed, the more it will cost you in expended energy. It would take an infinite amount of power to provide infinite acceleration to even a massless point charge if there were such a thing.
The speed of light (which doesn't have quite the same meaning as the speed of a photon) is a property of the vacuum of space, "inhabited" by electromagnetic fields described by Maxwell's equations. An electromagnetic wave (not quite the same thing as a photon) moves at a speed determined by the properties of the matter in space the wave is moving through, so "acceleration" of an EM wave (i.e., change of speed or direction) can always be traced directly to a change in material properties along the spread-out "path" of the wave.