There is definitely a difference in meaning between these phrases:
". . .where I understand he is doing a fine job," implies that you are familiar with the work he is doing there. Perhaps someone from his new company has told you he is doing well, or you keep up with the man in question, who then told you himself. Essentially, you have some form of (probably anecdotal) evidence about the situation — although you do not know for certain, or necessarily even believe it to be true yourself (you are taking it on someone's word, for example).
". . .where I know he is doing a fine job," implies that you don't have any information about his work. This is something one would expect to hear someone say of their relative (e.g. a mother talking about her son), a close friend, or perhaps a respected past-coworker. Essentially, you have no form of concrete evidence or information about the person's work and are instead trusting your personal intuition about the person's abilities. For example, "My son just switched jobs. He's been busy and we haven't talked, but I just know he's doing a fine job there."
In the case that you actually do know for certain that the man is doing a fine job (possibly because you've seen it yourself), the most natural phrasing would simply be to state so:
". . .where he is doing a fine job."
If you wanted to use the word "know" in this case, I recommend you somehow qualify your statement:
"I know he is doing fine work because I checked it myself."