While cats are not mentioned as domestic animals in the Bible, there are references to them in the Bible. Probably the most infamous "cat" story in the Bible is that of Daniel in the lion's den.
Although it is true that cats are not referenced as pets in the Bible, neither are any other animals. The Bible does not explore pet ownership. This could quite simply be because pet ownership was not crucial to the stories being told. The sparse mention of cats in the Bible may also be because cats were sacred to the Egyptians and the Egyptians tend to be vilified in the Bible.