It really depends what you mean by "rock". At the temperatures and pressures at the cores of stars (and at which nuclear fusion reactions are possible), "rocks" as I suspect you are thinking of, do not exist.
Thermonuclear reactions do not occur because the gas is "flammable", they occur because the kinetic energies of the nuclei in the gas (at these temperatures, atoms are fully ionised) are sufficient to get them close enough together for nuclear fusion to take place.
"Rocks" are made of atoms of silicon and oxygen (for example) in the form if silicates. But these are dissociated at fairly low temperatures compared to the centres of stars. Oxygen and Silicon thermonuclear fusion ignition requires temperatures in excess of $10^9$ K, and these temperatures are only reached late in the lives of stars of mass $>8 M_{\odot}$.
It is possible to have stars with solid cores. This is thought to be the fate of white dwarf stars as they cool. Most white dwarfs are made of a mixture of carbon and oxygen and this "crystallises" once the core of the white dwarf cools below about a few million degrees.
Ordinarily, the cores of white dwarfs are inert as far as nuclear reactions are concerned, because the temperatures are too cool. However, if the white dwarf is massive enough (or mass is added to it), then the central densities climb, and for white dwarfs of mass $\sim 1.38M_{\odot}$ it is thought that the densities become high enough ($\sim 10^{13}$ kg/m$^3$) to start nuclear fusion of carbon via the zero point energy oscillations in the crystalline material (so-called pyconuclear reactions). Such reactions in degenerate matter are highly explosive and might result in the detonation of the whole star in a type Ia supernova.