Could i connect my airpods to my apple watch and listen to music while

Yes but first you have to have music on your watch not just your phone. Check this out

Listen to music, podcasts, and audiobooks on your Apple Watch - Apple Support

Mort

Yes you can.

Use AirPods and other Bluetooth accessories with Apple Watch

1. Put your Bluetooth accessory in pairing mode using theinstructions that came with it.
2. On your Apple Watch, tap Settings > Bluetooth. Your Apple Watch will search for nearby Bluetooth accessories.
3. Choose your Bluetooth accessory.
4. If asked, enter a passkey or PIN.