How to stop the buzz sound when a message comes in?

Check if your settings are okay in:

Settings>Sounds&Haptics>Text Tone

and/or

Settings>Sounds&Haptics>Play Haptics in Ring Mode

and/or

Settings>Sounds&Haptics>Play Haptics in Silent Mode

and/or

Settings>Notifications>Messages>Sound

When done, restart your iPhone.

If those settings meet your needs, I can't imagine why you're hearing that buzz.

Try going to Settings>Sounds and then toggle Vibrate on Ring/ Vibrate on Silent. Hope it helps!