Precise knowledge of the total number of magnons, including both coherent and incoherent (e.g. thermal) excitations, is imperative for the advancement of fundamental spin-wave physics and the development of next-generation magnonic devices. In particular, quantifying magnons is key to understanding magnon transport phenomena, the nonlinear regime, or ultrafast magnetization dynamics. Typically, incoherent magnons are accessed by frequency-domain techniques, which lack the temporal resolution required for ultrafast processes, while ultrafast time-domain methods are generally sensitive only to the coherent dynamics. In this work, we demonstrate that femtosecond noise correlation spectroscopy enables a fully quantitative, time-domain measurement of both thermal and coherently excited magnon modes in bismuth-substituted yttrium iron garnet driven by a free-running microwave. We model the experimental data and extract the magnon number by simulating the magnon band structure of the sample, the magneto-optical response function, and the optical spot size used in the experiment. Our analysis establishes a connection between magnon mode calculations and experimentally accessible magnetic properties and fiducially reproduces the waveform and amplitude of the magneto-optical correlation signal for different experimental conditions. These results open a new pathway towards the optical tomography of magnon modes in non-linear or non-equilibrium conditions and can be readily extended to study ultrafast incoherent dynamics in other condensed matter systems.