probably the scenario is the belt is slightly loose, so it make a noise on start up. it goes away because after a few seconds of slipping the pullys catch up to each other and the noise goes away. the danger here is the new belt will get a glaze on it from slipping and be even more prone to do it. the other possibility is there is too much resistance to turning from the pump because of issues to the pump bearings. I'd loosen up the belt and see if the pump seems to turn by hand with no grinding or other obvious signs of distress. If it seems good, retension the belt to where it has the proper amount of tension and see what happens
Geoff