Your about two different issues here. I think your main question is whether debt collectors can go after your mother's Social Security disability income to collect the debt she owes. In most cases, for most types of debts (but not all), that income is protected. However, if that money ends up in a bank account with other funds it is possible it could be at risk.
The second issue is whether the collector has the right to collect this debt from your mother. That will depend on whether she actually owes the debt, the amount is correct, and whether the statute of limitations has expired, etc.
I would also really encourage you to set up a free consultation with a bankruptcy attorney with your mother. Explain to her that even if she doesn't want to file, the attorney can help her understand what property, income or assets maybe it risk if she can't pay this bill.
Please let us know how this turns out for your mother.