Don't send them anything unless there is a request in writing with a verifiable telephone number. First, I would check your mother's credit report to see if anything has been reported as delinquent. If there has been something, you can follow it up via the information provided you in the report. If something is incorrect, you can contact the credit reporting agency.
This approach by the "collection agency" seems to have the elements of a scam, so I would be very cautious.
I think I would need a lot more information about your mother. First, would it matter if her credit were damaged? If not, I really wouldn't worry about it unless someone else was on the account. I would send the collection agency a letter and tell them that you are demanding verification of the debt. Additionally, based on what you have provided, it is unclear as to whether they may have violated the Fair Debt Collection Practices Act. Who did the collection agency call? You or your mother? Do you live at your mother's address? A lot of this information would need to be known before I could provide you with more advice.