Like the brain, the receptors commonly called 'pain' receptors aren't in the spinal cord itself, but they are present in the meninges around the cord, and damage to the cord can be perceived as pain in particular because it can increase activity in the pathways carrying 'pain' information to the brain.
Nociceptors, often called pain receptors, are peripheral structures that sense and transmit noxious stimuli to the central nervous system. The cell bodies of afferents from these receptors are found in the dorsal root or trigeminal ganglion. The receptor structures themselves (the afferent terminal of these neurons) are not found in the parenchyma of any central structure, though they do innervate the meninges of both the brain and spinal cord. Despite the lack of peripheral receptors of nociception, damage to the spinal cord is often associated with increased activity in the tracts that carry nociceptive information.
This may be beyond the question you're asking, but I would note that the IASP defines pain as:
a sensory and emotional experience associated with actual or potential tissue damage or described in terms of such damage.
Pain is probably best understood less as a sense that nicely maps the external environment to a conscious representation of that environment, and more as an emotional need state, like hunger. Many different receptor structures and nerves can transmit a signal that can eventually be interpreted as pain, and in fact pain can be experienced without any physical peripheral input at all (as the IASP definition attests). This is one of the reasons nociceptors are typically not called pain receptors by people who treat or study pain.