My mom is in a nursing facility in NJ. Are they entitled to receive her pension?

If Medicaid is paying for her care the answer is yes. Since I knew my Mom would be in the NH for the rest of her life, I allowed the NH to become payees for her SS and pension.

As I understand as long as Medicaid paying for her, yes.

Is there a spouse at home who relies on her pension for support?

All her monthly income is subject to be used for her copay or SOC (share of cost) required by Medicaid.

If she has any recurring debts - like credit cards, life insurance policy payment, auto or property costs - all those she will default on unless family want to pay for them out of a sense of familial duty on her behalf.

Ye's, if it is needed to pay the bill for her care. Social Security can't be enough.