C210 Pilot with Flight Instructor reported NMAC with a C172.

Narrative

On Day 0; at approximately XA:26 PM; my Certified Flight Instructor (CFII) and I were flying from ZZZ to ZZZ1. We had recently departed from ZZZ Runway XX; setting up for the ILS approach to Runway YY at ZZZ1. My CFII was handling the radios while I was using a view-limiting device for practice purposes. At the time of the incident; we were located 4.4 nautical miles southeast of ZZZ over the VFR Checkpoint; short of the Bravo and Mode C Veil Shelf. After monitoring ZZZ's radios; we listened to ZZZ Approach. ZZZ Approach was vectoring us into ZZZ1 when they warned us about incoming traffic bearing at our altitude at 8 o'clock. My CFII took the controls and descended from approximately 2300 ft. to 1787 ft. MSL to avoid a Skyhawk possibly flown by aircraft owner: Person A. My CFII reported that the other aircraft was very close to us and made no attempt to alter its course. After the incident; we were cleared back to our altitude for the approach back into ZZZ1. We landed safely at ZZZ1 minutes later. Despite the close proximity of the other aircraft; they made no radio calls on ZZZ's frequency; made no radio calls on the ZZZ Approach frequency; and made no attempt to avoid our aircraft. We were able to later retrieve information on Aircraft Y. The aircraft has a record of its flight plans that evening; which later showed it arriving at ZZZ2. We discovered their information on the internet. I would like to express my gratitude to ZZZ Approach and my CFII for their efforts and quick action during this near-mid-air collision. If any authority figure would like to reach out; I can provide more information.