I think the way chivalry is represented in "Gawain and the Green Knight" was one which was constructed by "court nobility," meaning that there are a variety of rankings within the noble class. There is often violence associated with chivalry in the noble class system; however, the violence is heavily contrasted by the Christian undertones of King Arthur and his court, as well as the popularity of Christianity during the Medieval Period. For example, the initial beheading takes place during Christmas festivities showing that the knights and King Arthur are Christian and celebrate/practice the Christian faith. Normally, one would think of this beheading as something unchristian like and not chivalrous; however, it is part of knighthood and being a knight of King Arthur's Court.
We also see chivalry play a role during Gawain's interaction with the wife of Berilak. She is coming at him, like Guinevere advanced towards Lanval, and he is dodging her advances. There are some parallels with Lanval during this part of the story. Gawain cannot accept her because he will be less honorable; however, he is conflicted because he cannot downright refuse her or he will make her angry. Gawain is walking a very very fine line between upsetting his hostess and dishonoring himself.
Overall, chivalry is a little different than how we have seen it before. We see the beheading game that is occurring on Christmas, muddling the lines between chivalry and not sticking to the moral code of Christianity. There are also parallels, however, that we see in the interactions with Gawain and the wife. This further proves that chivalry is not just one thing and can be expressed and shown in different ways.