I have a similar noise but it seems to come from dashboard, when car cold a whirr is heard right after start
and the clock and the trip meter zeroes. I might be that my battery is getting weaker (especially when cold).
Does your clock and trip meter go to zero when it happens ?
If it comes from the dash board then like the above posts say, could be a relay. I dont know tbh, but are your relays in this old car in the dash board bear ?