As long as you are positive that you did not sign anything at the nursing home listing you as financially responsible for your mother's nursing home bills, then no, you do not have to pay them. However, if your mother left an estate, the nursing home could go after any assets included in the estate for the bills, but not you personally.
Nursing homes are famously dense when it comes to billing and debt collection. My mom once stayed several weeks in a facility where she received excellent medical care but really poor financial treatment. It took months to resolve and they kept sending the bills to me. What a mistake that was! They finally resolved matters with her insurance company and she even got her co-pays back.
I recommend that you contact the collection company and request verification that the debt belongs to you and that you are the responsible party. Because you did not sign anything at the nursing home, the collection company will not have anything to send you as verification. Wait 30 to 45 days, and then I recommend that you send the collector a letter, stating the debt is not yours and you do not wish to be contacted regarding this attempted debt collection again. Send the letter certified mail with a return receipt request.
The Fair Debt Collection Practices Act requires that the collector honor your request and not contact you again regarding this debt. Should the company violate the law and contact you again, you can file a complaint with the Federal Trade Commission at FTC.gov.
I also suggest that you send a copy of the certified letter to the nursing home. I'd add a cover letter to the nursing home that says if it sells or places the debt with another company, you'll consider legal action against it and report it to the local regulators in that state.
To assure this does not negatively affect your credit, check your credit report with each of the three major credit bureaus at AnnualCreditReport.com and see if the nursing home or collector has reported the debt to the credit bureaus. Chances are, the nursing home won't report the issue, but the collector might. If it is reported, dispute the item with any and all of the credit bureaus that are reporting it. Check the bureaus' websites for information on how to file a dispute. All three major bureaus provide for filing your dispute online.
Once the dispute is filed, the collector will be contacted by the bureau and will have to prove that the debt is legitimate in order for it to remain on your reports, and they don't have any proof. The bureaus have an interest in keeping their data as accurate as possible because they sell the information to lenders and others who rely on their data integrity. The bureaus will send you a report on their findings regarding the dispute and should let you know that the item has been removed. Be sure to check your credit reports in a few months to ensure that the item actually has been removed.