If I buy an Apple Watch 8 w/cellular will my Verizon bill increase?

Set up cellular on Apple Watch - Apple Support

Contact Verizon and ask them. It has to be a supported plan.

Make sure that you have an eligible cellular service plan with a supported carrier.

Only if you add watch to your plan for it to get cellular service when away from your iPhone.

If you don't activate cellular service on it, then it will work as a GPS only Watch and you will not get charged extra for it.

If you do add service to it, your carrier will usually add a service fee to your iPhone plan for it. Around $10 in most cases in the U.S.