If you are the suspect in a new crime and are asked by the probation officer about that new crime, you can assert the Fifth Amendment. The fundamental question is whether and when you can face probation violation proceedings for refusing to answer the questions on that basis.
I would argue that the valid assertion of the Fifth Amendment should not form the basis of a probation violation, but that the defendant remains at significant risk of being found to have committed a probation violation on other grounds. Terms of probation frequently involve provisions that a probationer may not get arrested, or even to more broadly be involved in an encounter with law enforcement, so it's possible that the circumstances of the new charge would support a probation violation even without more evidence of the probationer's actual involvement in criminal activity. Also, a probation violation proceeding is a probable cause hearing, and it will often be possible to meet that standard of proof even if there has been no conviction on a new charge, and even if charges haven't been filed on a new, alleged offense.

