Nothing is completely impossible, but this is as close to impossible as it gets.
The S.M.A.R.T. system, both the testing program and the tracked data, are stored on the disk's controller. The motherboard is literally just passing the data that the hard drive reports to it. You should be able to move the disk from one system to another and retrieve the exact same S.M.A.R.T. data.
Furthermore, it should also be impossible for read/write requests from the motherboard to cause the disk to falsely detect bad or damaged or potentially damaged sectors on the hard drive. Again, the motherboard is just passing requests to the hard drive, and the hard drive is responsible for operating in a manner that doesn't cause itself to be damaged.
The thing to remember is that hard drives are small computers today. They're not dumb electronics. They won't break if you tell them to read sector -1 (very early ones would!). They contain software that validates requests from the attached computer before trying to do the disk operations. The fact that software is present to do this on the hard drive is why hard drives now often have firmware updates.
It's possible that the firmware on the hard drive was faulty -- hence firmware updates -- but, again, the problem would still be with the firmware and not the motherboard. Even if you do a firmware update that fixes a known issue, however, it would very likely not recover or reset any of the blocks marked as bad. Those are essentially permanently marked as unusable.
It is also possible that the motherboard is broken as well, but it's not likely to be broken instead of the disk. Hard drives just don't work that way anymore.