What determines the intensity of light in Quantum theory?

The level of excitation.

the number of photons

The density of quanta (quanta/cm^3).

Light is made up of particles called photons, and photons are emitted by energetic electrons as they fall from higher to lower “orbits” (states of energy) inside the atom, so the intensity of light is determined by how much energy the electrons have.
It’s not just the number of photons given off, but also their wavelength, that determines the intensity. Infra-red (heat) is a low energy emission. Ultra-violet is higher energy. The larger the jump in “orbit” that an electron makes, the higher the amount of energy in the photon that is given off.
Visible light, infra-red, ultraviolet, microwaves, radio waves, X-rays, they are all part of the electromagnetic spectrum, and they are all carried by photons with different wavelengths.