In the early series, the Cat said that he didn't have his own quarters, nor did he want them. However, by Series X the Cat had acquired his own bunkroom.
It could be assumed that, given he was not a registered crewmember, Holly never assigned the Cat his own quarters. In the first series, the Cat would often simply disappear into Supply Pipe 28 when not hanging around with the "monkeys" (what the Cat calls humans). It was in Supply Pipe 28 that the Cat was presumably raised by the Cat Priest.
Presumably the reason why the Cat didn't have assigned sleeping quarters was, like his feline ancestors, he would sleep where he wanted to. This was evidenced when at one point, Rimmer moved out of the main sleeping quarters to live with his hologram clone. Lister, now living alone, asked the Cat to move in with him, but he said that he preferred to not live in any one particular place, and would rather just bunk down wherever he felt like. ("Me²", Series I)
Sometimes the Cat could be seen sleeping in Lister's bunk when Lister was not in it, such as in "Thanks for the Memory".
Rimmer once caught the Cat making numerous photocopies of his rear end. When Rimmer asked what he was doing, the Cat said that he was making wallpaper and matching drapes for his room. This is the first mention of the Cat having his own quarters. ("Dear Dave" deleted scene, Series X)
The Cat's quarters are first seen when he has a dream of being in bed with two female Felis sapiens. Much like his clothes, the general theme of the Cat's bunk was that of bright, flashy and garish colours. ("Can of Worms", Series XI)