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.

Example:

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.

Example:

When the issue became contentious, the negotiation had to be halted until both parties regained their composure.