Physics isn't my strong point but the moon is showing for another reason not due to heat as it's a huge distance away and isn't warm.
Of course it is warm. It's a piece of rock being shone upon by the Sun, it' at least as warm as any piece of rock around you during the day and those shine bright and strong on the IR vision. In fact, due to the lack of atmosphere, the surface gets actually much warmer than that in some places - even over 100 Celsius actually - because there is no air to cool the surface down and the very porous surface is also very good thermal insulation, so all the heat received stays close to the surface. Additionally, it also reflects the Sun's IR radiation, but I would make a mildly educated guess that that's a minor effect.