Connection airpod to apple watch

Yes, the AirPods work with the Apple Watch even when the iPhone is not nearby.

If you are having difficulty pairing the AirPods with the Apple Watch, try pushing the little (hard to see) button on the back of the AirPod case to put the AirPods into "Discover" mode. Then see if your Apple Watch -> Settings -> Bluetooth can see them.

Take a look at the many links in AirPods Support - Official Apple Support which might have the clue you need. AirPods should work without your iPhone nearby, but you might have to fight an obstinate device to get the initial pairing to the watch.