My cleaning people accidentally locked my cats in my bedroom and they subsequently peed on my bed, ruining the mattress and pillows. Will my renters insurance cover this kind of pet related damage? I use State Farm.

probably not, but ask the cleaning people to pay for damages that is why they carry insurance

No. The standard renters policy has an exclusion, to cover any damage to a premises you rent. There's ALSO an exclusion for damage to your property by your pets.

Regarding the contents, the bed and mattress, most policies cover on a NAMED PERILS basis (including fire, theft, vandalism, smoke, wind, hail, etc), and "my cat peed on it" is not a named peril.

Awwwww. We wish it did but...nope. If you read the policy you will find a section called "exclusions". Damage by pets is ALWAYS excluded.

We buy insurance to cover financial exposure/expenses that we ourselves can not cover. Cat accidents are typically within our ability to pay. Plus - you have a deductible anyway.

No sorry, damages your own pet does to your own stuff is not covered. It is specifically excluded in your renters policy. You can TRY to go after your cleaning people's general liability policy but you would have to prove negligence. Did they KNOW the cats were in the room when they closed the door? Cats are notorious for hiding so they may not have known they were in there. Good luck to you.
When you get a new mattress, get a good mattress cover (they are waterproof & terrycloth on top) and do not feel funny. You don't even know it is there. Your local mattress store should have them (they are about $65 but worth it - sorry I don't remember the brand we got). They make them for pillows too. If your new mattress has a warranty, they may require you get a good mattress cover. I have one for my son who had accidents when he was younger & when we got our new mattress I got one for us too because I was pregnant.
It also keeps dust & dust mites from getting into the mattress.

No, the cleaning people should pay to have your mattresses professionally cleaned and to replace pillows.