No, Contentious is not derived from content. It’s not its adjective. It’s not another noun with the same meaning. It’s a totally different word that has a totally different meaning.
Content [kuhn-tent] or Contented [kuhn-ten-tid] (adj.):
satisfied, contempt, agreeing.
You can’t bribe me with your money, I’m contented with my own.
Contentious [kuhn-ten-shuhs] (adj.):
1. argumentative, quarrelsome.
2. causing argument or strife.
When the issue became contentious, the negotiation had to be halted until both parties regained their composure.