Not necessarily. As mentioned these are probably points where the tooth is attempting to push through the gum. You would know if it was infected. Your gums would be red and swollen and extremely painful.
Maybe - Maybe not...
From my recollection mine was not infected. The white spots near mine was actually a thin areas of the gum that was not ready to erupt or cut through the gum.
My Oral Surgeon said it must have been painful eating things like chips or pretzels of which I wasn't able to tolerate.