Apple watch

The Apple Watches are not locked to a carrier. When you activate the cellular on a watch, it is done with your existing carrier and it creates a clone of your iPhone SIM to work on the Watch, since it uses your number from the iPhone. The current cellular account from AT&T would have to be removed from the Watch and then if you wish to activate cellular on the phone, activate with your Verizon account.

Is this a Watch you would be purchasing from someone else? If so, make sure they perform all of the steps listed here, What to do before you sell, give away, or trade in your Apple Watch, or buy one from someone else - Apple Support

Did AT&T require you to activate this Watch when you purchased it? If so, where did you get an iPhone with AT&T to activate it with? If they did not require you to activate it with them, then there is no issue for you to activate with Verizon, unless they refuse since you did not purchase it from them. But since you are a customer with your iPhone, I wouldn't see a problem.