It’s a long way from the Crusades to the modern sense of the word, but that’s the path we must follow to explain where we got termagant from.
In the Middle Ages, Crusaders knew almost nothing about Islam and lumped everyone non-Christian together as Saracens or pagans; they thought that Mahmound (or Mahound) and Termagant were among Islam’s gods. The word appears about 1100 as tervagaunt; that was in French, in the Chanson de Roland. The Italian equivalent was Trivagante, which may in turn have come from Latin words that meant a threefold wanderer; this referred to the moon, which was considered to travel between heaven, earth, and hell under the three names Selene, Artemis, and Persephone.
The word was borrowed into English and became the usual name in the medieval morality plays for an overbearing, violent and turbulent character, the supposed god of the Saracens, who was always dressed in Eastern robes. He was borrowed by Shakespeare and put into the mouth of that early theatre critic, Hamlet: “I could have such a Fellow whipt for o’erdoing Termagant: it out-Herods Herod.” The name had by then been generalised to mean a quarrelsome person or bully.
Around the middle of the seventeenth century, the word changed sex to become our modern term for a quarrelsome woman. It’s likely that people were confused by the Saracen’s robes in the morality plays and assumed that the supposed god Termagant was really female.