Swipe up on the Apple Watch face to open Control Center
Tap the Silent Mode button
Wait for the bell icon with a line through it to appear
Open the Settings app on your Apple Watch
Tap Sounds & Haptics
Turn off Haptic Alerts if you want no vibration
Turn the alert volume down to the lowest level if needed
Use Do Not Disturb from Control Center to silence alerts temporarily
Enable Theater Mode from Control Center to mute sounds and prevent screen wake-ups
Check that Focus modes are not allowing notifications to make noise