A frequentative form (that is, one denoting repeated action) evolved, argutāre; this signified ‘make repeated assertions or accusations’. This passed into medieval French as arguer ‘accuse, blame’, and also ‘bring forward reasons for an assertion’, and thence into English. The meaning ‘accuse’ died out in English in the late 17th century, leaving ‘reasoning, discussing’ as the main sense of argue.
Meanwhile, original Latin arguere had made its presence felt in establishing the sense ‘prove’ in English, now somewhat weakened to ‘imply, indicate’ (as in ‘Their lack of involvement argues indifference’). The sense ‘quarrel’ seems to have developed from ‘discuss’ in the 17th century.