I have been living in the same apt for 3 and a half years and recently was told I had to move out with no reason stated. As we were packing my 3 yr old daughter noticed something hanging from the ceiling it looked like a spider web bt really thick, I didn't know that was signs of termites, as we continued to pack my husband noticed on our bed a book opened and covered in termites, as we looked around to see if they were anywhere else they had went into everything, almost everything is ruined, I know they have left notices to spray but they never entered my apt or came to it at all. Is my landlord resposible for the damages to property? What can I do ?

You can take pictures and possibly save some evidence in a very strong plastic container. And then sue the landlord.
Termites live in and eat wood. Most of your possessions should be safe. What is ruined?
Exterminators wouldn't enter your apartment to treat termites. They treat the foundation. But if there were a lot of them, they may have come upstairs. Yes, they do leave strands of loose fluffy stringy stuff, and wood powder.
I would put boxes of things open outside for several hours. Try spraying everything with something obnoxious but not harmful to people, just to drive them away, like ammonia based cleaners. Unlike cockroaches, they don't want to live in your possessions.