Look at a few things first
Look at a few things and see what you come up with. Seeing as most all water pumps have sealed bearings, the seals go before the bearings do. If the seals are actually going out on your pump, then it should be spitting coolant from the spit hole located on top of the water pump housing, and if that's covered up by the harmonic balancer at all, you'll have to feel around the hole after the engine is shut down for a wet spot. The other item, and one that most folks overlook is the belt that drives the pump. Although it appears fine and dandy, it may either be slipping or has glazed over the contact surface enough to cause this noise during start up then go away after the belt has flexed enough to become limber again. Both are cheap alternatives to troubleshoot or repair, and if in fact it is your pump that is leaking and grinding, do it yourself and save the bucks on labor, the part can't be that pricey.