This is interesting because I have a similar issue. I'm not sure it's a "problem", because the hub (Alfine 11) works perfectly. It has almost 9,000 miles on it, and has been well-maintained (oil changes every 700-1,000 miles). Ever since the last oil change (approximately), the hub has been making a soft "chirp" once per revolution. I've checked fenders and stays, and brake alignment and ruled those out. The sound is actually a bi-modal sort of thing, more like "CHIRP chirp", the frequency of which does *not* change with wheel speed, i.e., the rate varies with wheel speed, but the pitch of the chirp is constant. Also, like your noise, it only occurs when I am seated and riding the bike. This would seem to point toward the tire, but there is clearance everywhere. If I pump the tire up to the maximum pressure (60 psi), the sound is markedly reduced... again, suggesting the tire. Aside from verifying the clearances, I would think that a tire contact would change its pitch with increasing speed. So I'm mildly puzzled. Could this be a spoke tension thing? The spokes all "ping" about the same when tapped with a screwdriver. I will eventually have to try removing the fenders completely, and checking the spoke tension quantitatively. Anyway, FWIW!

BTW, the Alfine 11 marks align in gear 6, not gear 4. (Post #2)
Of course, with an Alfine 11 that's "new-to-you", you should change the oil so you can rule that out as a potential issue.
EDIT: I guess it's not new to you, but if you haven't changed the oil for a while, that would be a good place to start. FWIW, I've been using Royal Purple "MaxGear" gear oil, 75W-140. This oil is very similar to the "official" Shimano oil, but much less expensive. 75W-90 would work as well.