No. Answer over.
Just kidding. The Germans did not employ a shotgun during World War II, and neither did most countries at anything beyond a small scale. Shotguns were used during World War I and have come back into fashion recently because of the high firepower they can deliver in close-quarter combat.
In World War I, they could deliver a massive manstopping power against enemy infantry and skirmishers raiding your entrenched positions or, inversely, deliver the same in the close quarters of the enemy trench. On the flipside, modern warfare's shotgun units profit from the fact that they can be used fired blindly or at low visibility into corridors, around corners or into closed rooms with overwhelming damage. However, such usage generally is done by a specifically trained shotgun-trained soldier, which is a point we will get to.
The war of medium distance combat
World War II meanwhile was a war that was at least conceptually in the minds of most arms suppliers and military planners initially supposed to be fought at some distance by the infantry, which is where the mostly reliable, mostly precise and cheap-to-produce semi-automatic and bolt-action rifles shine. When indeed World War II became a war of close quarter action, mostly in the Häuserkampf of the Eastern Front between house-clearing parties on one side fighting desperate defenders entrenched in the houses of the other, the shotgun's responsibilities were left to its cousin, the hand grenade.
The hand grenade as the alternative to the shotgun
The hand grenade too can be deployed in small corridors, around corners and into closed rooms, and will in fact do more damage to the enemy than a directional shotgun blast would - at the detriment of potentially inflicting more friendly fire. Furthermore, hand grenades were cheaper to mass produce and could be delivered, at least in theory, to all frontline infantry troops rather than to just dedicated shotgun bearers.
Shotgun service in other countries, particularly the USA, and its World War I context
That said, some countries still delivered shotguns to their units to some degree. The 1897 Winchester, already successful in World War I, saw service with U.S. Armed Forces in World War II. The Winchester's usage (in its short-barrel form known as the "Trench Gun") had actually been subject of a diplomatic protest by the German Empire in World War I, as the Germans felt that shotguns violated the terms of war.
Even back in the First World War, the German military preferred using fully automatic weapons like the Maschinenpistole 18 over the slower-firing type of shotguns deployed by the Entente (the United States in particular, as the other two major western Entente powers did not field significant numbers of shotgun shock troops).
On that note, usage of shotguns even by American troops was rare in Europe. The Winchester did not have its own designation as a potential Beutewaffe, a scavenged weapon, unlike the M1 Garand and M1 Carbine, which the Germans upon capture referred to as Selbstladegewehr 251(a) and Selbstladekarabiner 255(a), respectively.
The developmental direction of German infantry armament
So there is probably a certain German military culture with aversity to the usage of shotguns that also needs to be considered. The progression of German troops armament was to generally enable the German infantryman to deploy more shots, advancing from the bolt-action Kar98 over the semi-automatic G41 and G43 to the fully-automatic MP40 and Stg44. To add shotgun roles or even dedicated shotgun shock troops would have gone against this trend. The German leadership instead preferred to expand the "grenadier" functionality of its infantry units and thus left the combat role fulfilled by the shotgun to the hand grenade.
So the answer is "no". The reasons for the lack of shotgun usage are as follows:
Infantry was expected to mostly engage the enemy at some distance where rifles were preferential.
In close combat situations, the infantry was already sufficiently equipped with hand grenades to not necessitate the specific training of shotgun experts.
German military culture, next to a long history of 'grenadier' units and the favorable view it already had upon hand grenades, had been going into the direction of rapid fire weapons as the preferred choice for infantry weapons. The "evolution" of the infantryman was towards the fully automatic machine pistol and assault rifle, not the slow-firing shotguns.