Is there any evidence for the story of the king forbidding intercourse between men and women in the time of Abraham (peace be upon him), and Abraham's mother conceiving him under the king's throne, this being presented as proof for quarantine during the coronavirus pandemic?

The story of Ibrahim's birth, peace be upon him, mentioned in the question, is from the Israʼiliyyat (Judeo-Christian traditions) and is not established in the Qur’an or Sunnah (definition). Books of Tafsir (definition) (exegesis) mention that the priests and astrologers informed Nimrud that a boy would be born who would destroy his kingdom, so he ordered the slaughter of every newborn male and separated men from women. However, Ibrahim’s parents, peace be upon him, met and his mother conceived him under exceptional circumstances.

As for linking this story to quarantine, it is an inaccurate inference. While it is true that means do not act independently and that Allah's will is paramount, this does not mean abandoning legitimate precautionary measures. Quarantine is a necessary measure for preventing epidemics. Reliance on Allah does not mean abandoning means, but rather employing them while depending on Him.