Nearly all thermal radiation phenomena involving materials with linear response can be accurately described via semi-classical theories of light. Here, we go beyond these traditional paradigms to study a nonlinear system which, as we show, necessarily requires quantum theory of damping. Specifically, we analyze thermal radiation from a resonant system containing a $chi^{(2)}$ nonlinear medium and supporting resonances at frequencies $omega_1$ and $omega_2approx 2omega_1$, where both resonators are driven only by intrinsic thermal fluctuations. Within our quantum formalism, we reveal new possibilities for shaping the thermal radiation. We show that the resonantly enhanced nonlinear interaction allows frequency-selective enhancement of thermal emission through upconversion, surpassing the well-known blackbody limits associated with linear media. Surprisingly, we also find that the emitted thermal light exhibits non-trivial statistics ($g^{(2)}(0) eq 2$) and biphoton intensity correlations (at two distinct frequencies). We highlight that these features can be observed in the near future by heating a properly designed nonlinear system, without the need for any external signal. Our work motivates new interdisciplinary inquiries combining the fields of nonlinear photonics, quantum optics and thermal science.
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