Vatican
1550s, from Latin mons Vaticanus, Roman hill on which Papal palace stands. By Klein's sources said to be an Etruscan loan-word and unrelated to vates "soothsayer, prophet, seer" (see vates), but most others seem to think it is related, on the notion of "hill of prophecy" (compare vaticinatio "a foretelling, soothsaying, prophesying," vaticinari "to foretell").
vates (n.)
1620s, "poet or bard," specifically "Celtic divinely inspired poet" (1728), from Latin vates "sooth-sayer, prophet, seer," from a Celtic source akin to Old Irish faith "poet," Welsh gwawd "poem," from PIE root *wet- (1) "to blow; inspire, spiritually arouse" (source also of Old English wod "mad, frenzied," god-name Woden; see wood (adj.)). Hence vaticination "oracular prediction" (c. 1600).
Quirinal
royal palace in Rome, later the Italian presidential palace, 1838, from Mons Quirinalis in Rome (one of the seven hills, site of a former Papal palace), from Quirinus, said to be the divine name of Romulus, but rather one of the original trinity of Roman gods, assimilated to Mars. His feast (Quirinalia) was Feb. 17, the day Romulus was said to have been translated to heaven. Used metonymically for "the Italian civil government" (1917), especially as distinguished from the Vatican.
https://www.etymonline.com/word/Vatican#etymonline_v_4661