In the early series, Cat said that he didn't have his own quarters, nor did he want them. However, by Series X Cat had acquired his own bunkroom.
It could be assumed that, given he was not a registered crewmember, Holly never assigned Cat his own quarters. In the first series, Cat would often simply disappear into Supply Pipe 28 when not hanging around with the monkeys (what Cat calls humans). It was in Supply Pipe 28 that Cat was presumably raised by the Cat Priest.
At one point, Rimmer moved out of the main sleeping quarters to live with his hologram clone. Lister, now living alone, asks Cat to move in with him - but Cat 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 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 Cat making numerous photocopies of his rear end. When Rimmer asked what he was doing, Cat said that he was making wallpaper and matching drapes for his room. This is the first mention of 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 Cat's bunk was that of bright, flashy and garish colours. (Can of Worms, Series XI)