Assuming we are talking about regular, non fluorescent objects, then yes, the perception of color on objects comes from the reflected light in most cases. The light that is emitted by an object is not the same wavelength as the one that was absorbed, and for the temperatures we are used with, this radiation is not even in the visible spectrum.
When light is incident on a surface part of it will be reflected, that is the light that gives rise to the color of objects. The other part will be absorbed. The absorbed part will not be re emitted in general, but in fact it will usually cause a lattice excitation that dissipates through the solid, which can be translated as the solid increasing its internal energy, i.e. increasing its temperature. This energy will be released via thermal radiation, called blackbody radiation, and for everyday temperatures the emitted radiation is not on the visible espectrum, so it would not affect the color perception of the object.
An example of color coming from blackbody radiation is the glow from an incandescent object. It is easily seen from heating up metals, that as they become hot they start glowing red.