Who is responsible for the debts on a property after a quitclaim deed has been filed and signed?

If by "debts" you mean the mortgages on the property, the borrower remains liable for the mortgage debt AND the new owner takes the property subject to the debt.  A successful foreclosing lender can take title to the property and also pursue the original borrower for any deficiency. Also, there are very few instances where a quit claim deed is the appropriate instrument to use to transfer title to property. Typically some form of a warranty deed is preferred so as not to negate the warranties of title from any existing owner's title insurance policy. The foregoing is specific to the law and procedure in Florida.  This response does not constitute legal advice as the facts presented are limited and unstated facts will likely impact your particular situation. This response is intended for general education only and does not create an attorney-client relationship. Please schedule a consultation with a local attorney for more specific and detailed answers to your legal issues.

It depends on the debt.  Some might be personally obligations relating to the property, such as a utility bill that was created by the first owner.  The utility company will continue to look to the first owner for payment.  Of course the first owner should try to get reimbursed by the second owner, or shut off the utilities.  Usually an agreement  is entered into.  Also just the signing of the quitclaim deed doesn't create an obligation in the second owner; it has to be accepted.  And even then it is advisable to have an agreement that outlines the responsibilities of the parties after the delivery of the deed.  So it would help to analyze specific details of the debt and the transaction.