Do cats produce DMT?

Yes! Many mammals, including cats and humans, produce DMT in small amounts.

DMT has been detected in blood and urine and is produced by the enzyme indolethylamine N‐methyltransferase (INMT). These do bind to serotonin receptors. Serotonin receptors are actually expressed in multiple tissues outside of the brain. However, DMT does cross the blood-brain barrier (ref), so it's possible that endogenous DMT has some neuroactive effect (I really don't know). Good question.

This 2016 paper states (referring to a 2012 review) that it was identified in urine, blood, and CSF:

A review by Barker in 2012 assessed 69 studies that reported endogenous DMT detection and quantities reported in urine (29 studies), blood (11 studies), and cerebrospinal fluid (4 studies) from 1955 – 2010 primarily comparing detection levels within healthy controls and schizophrenic patients. DMT in urine was examined in 861 individuals (635 patients), 276 patients and 145 controls were positive for DMT.

I think that's the most direct answer to you question I could find. I didn't find anything on cats specifically, but that's likely to be similar.

This earlier 2004 paper states that it was only found in urine, but the previous study refers to studies before 2004, so I don't know why these two seem to contradict:

In mammals, endogenous bufotenine and DMT have been identified only in human urine. The DMIAs bind effectively to 5HT receptors and their administration causes a variety of autonomic effects, which may reflect their actual physiological function. Endogenous levels of bufotenine and DMT in blood and a number of animal and human tissues were determined using highly sensitive and specific quantitative mass spectrometric techniques.

Here's the entry for INMT on Kegg, and the Human Protein Atlas lists where INMT is expressed. Weirdly, that enzyme is highly expressed in the human lung.