Well I'd be dubious about that quote as we find a few different stories in which Genghis Khan apparently proclaims himself the wrath of god to various different groups, so maybe there was a grain of truth but it strikes me as apocryphal. However, there is a fascinating history of Mongol interactions with the western Christendom.

Christendom's first knowledge of the Mongols comes only in the form of garbled rumours linked to their passing sweep through Russia in the 1220s and their later campaigns in the Middle East in the 1230s which were incorporated into established tales of great Christian kings to the East in particular the Prester John legend. For the Church and the Christian world Asia is a vast mysterious terrain of which they have little concrete knowledge.

Concrete knowledge of the Mongols arrived in the singularly unpleasant form of an invasion of Hungary in 1236 which ended in 1242, likely due to the death of the Great Khan Ogodei which necessitated electing a new one and a light civil war. However, most of Christendom still had little idea with what exactly they were dealing with as speculation among the clergy ran from them being the ten lost tribes of Israel, which sparked a massacre of Jews in Germany, to it somehow being an elaborate ploy by the Holy Roman Emperor Frederick II to gain the submission of the King of Hungary to him.

Nevertheless even after the armies withdrew Church desire for more concrete information spurred Innocent IV to eventually send three embassies to Mongol territory, two Dominican and one Franciscan as these mendicant orders already had experience with missionary work in the Middle East. The two Dominican missions don't manage to achieve much one getting sidetracked by its other mission of trying to bring schismatic churches of the east back into the fold and getting no further than northern Iran, and the other while managing to reach the camp of the Mongol commander Baiju in western Asia all they manage to do there is offend him. Only the Franciscan mission under Plano of Caprini reaches Mongolia and is able to bring back some detail. However, this is not very encouraging while it allows an assessment of Mongol military capacity the prospects of deterring or converting are far from encouraging. A letter sent by the new Great Khan Guyuk telling the Pope and the princes of Europe travel to the Mongol court and submit himself or face annihilation. However, the justification we find here is not a mocking claim to be the wrath of god but the growing Mongol beleif that they were divinely destined to rule the world and that anyone who had yet to submit to them was a rebel against the celestial order.

The nest major mission is that of William of Rubruck who undertook the journey of his own accord to perform missionary work which lasted from 1253-1255. Here we get a fascinating account of his time there struggling to comprehend Mongol willingness to patronise all religions and even found himself drawn into a debate organised by the Great Khan Mongke between representatives of Christianity, Buddhism, and Islam which eventually collapsed into acrimony as the participants got drunk. Eventually William of Rubruck was sent back carrying a message explaining that the Great Khan belonged to no one religion as all were expression of one eternal god in heaven, this was then coupled with the claim that there was then one lord on earth and the usual calls to submit or be destroyed.

Things don't move much beyond this until the Mongol Empire fractures into various successor states at which point we have some interesting diplomatic contacts. Namely the possibility of cooperation between the Ilkhans and a crusader force against the Mamluks an idea which persists from the 1260s all the way until the 1320s. Alongside this were of course hopes of conversion encouraged by the presence of a significant number of Nestorian Christians among the Mongols. However, even after the Ilkhans definitively convert to Islam this doesn't seem to have quite registered with Europeans with various references to cooperation against abominable Mohhamadeans still present in the message from King Edward II of England to the Muslim Oljeitu in 1307. Still for Illhans looking for a strategic ally there was no reason too make too much of this.