Final answer:
King Henry II quarreled with Thomas Becket and was the father to both King Richard the Lionheart and King John.
Step-by-step explanation:
The king who quarreled with Thomas Becket and fathered two kings was King Henry II. King Henry II's dispute with Thomas Becket arose primarily from issues of church versus royal authority. Henry II, as part of his efforts to consolidate legal power, wanted clerical members who committed secular crimes to be tried in royal courts. Thomas Becket, the Archbishop of Canterbury, opposed this, leading to a bitter dispute that culminated in Becket's murder in 1170. Furthermore, King Henry II was the father of both King Richard the Lionheart and King John, who succeeded him and are well-known figures in English royal history.