My dad rec’d his SS check in January, but deposited it. He died later that month. Mom wants to deposit it into joint trust account. Can she?

The Social Security Administration website states:
https://www.ssa.gov/planners/survivors/ifyou.html

If the deceased was receiving Social Security benefits, you must return the benefit received for the month of death and any later months.
For example, if the person died in July, you must return the benefits paid in August. How you return the benefits depends on how the deceased received benefits:
For funds received by direct deposit, contact the bank or other financial institution. Request that any funds received for the month of death or later be returned to Social Security.
Benefits received by check must be returned to Social Security as soon as possible. Do not cash any checks received for the month in which the person dies or later.

I agree with JoAnn that your Mom can do what she wants with the January check because your Dad received it while he was alive.

My Mom died September 17th so I did not have to repay the SS check that was direct deposited by SS on August 31st.  The SS did not deposit a check on September 30th as that was after Mom had died.

If the check was dated January, its really Decembers. If he died in January, he can keep that check. If he received one February, that is January's the month of his death. That check will be requested back.

Was Mom on the acct check was deposited? I see no problem in her doing what she wants.