Yes you need to have the US side update your name. You also need to have your permanent resident card updated. The problem is the programs link to each other.

You can indeed use Nexus, your card didn't become invalid because of the name change. However to update the name on it, it must be updated in the US since I'm guessing that is where you applied and where issued the Nexus card. While the program is a combined effort, only the country that issues your Nexus card can update it.