When I acellerate hard, I hear a series of three or four clicks or pops from under my 2005 Dodge Durango. Could it be my transfer case is going out and need replacing?

I would inspect the joints in driveline.

it could be the rods or valves or timing chain or belt

had it, replace your 4 way universal, should be about in middle of drive line, if bad, when going up a slight grade,at a normal speed let off the gas then hit the gas again, try that a couple of times , if its bad you will hear the noise

a pop or click is not enough information,

Ok, lets just say it is loud clicks that are so deep they could sould like pops.