To answers your questions
1) The Apt Management Company says we have to pay for damages is it true?
The Answer is NO. Pest control is a function of management. Upon written (preferable certified mail) notification, the owner/manager should take steps to correct the problem. If they choose not to fix it, you have the option of going to the magistrate office and asking for a constructive evection (Google Constructive Eviction), bring evidence of infestation, the letter requesting the problem be fixed and an written response for the owner/managers.
2) Could the bugs be coming from another apartment? Absolutely. Bed bugs like and other pest migrate from one apartment to another. If you have never had a problem with bed bug prior to now.. It is more than likely you may have gotten them from an adjoining unit.
3) Paying for Damages? I would assume that the owner/manager is referring to the cost of Pest Control. I would ask for clarification. Bed bugs are no joke and difficult to get rid of. Good luck in your endeavors and I hope this helps. Please see enclosed link. If you need further assistance give me a call.
Century 21 American Properties, Inc.