Do you need upper body strength to rock climb?

You actually don't need prior upper body strength to start climbing. You will gain that strength as you climb.
Many people believe that to climb, you need to pull your whole body's weight with your arms, essentially doing pull-ups to get to the top of the route. This, though, is very far from the truth.

Think of rock climbing as climbing a ladder. You would not just pull yourself up a ladder with your hands, right? No, you would use your legs to push yourself up, and your hands to slightly pull, but mainly to balance/navigate yourself. The same logic applies to climbing, especially in the easier routes.

Beginner climbing routes tend to have large holds that are easy to grab on to and hold. These routes are essentially ladders, they usually go straight up. All you need to do is to climb them just like you would a ladder. Their purpose is to slowly get people used to the feeling of climbing, the shoes, and the holds. As the routes get harder, the holds get smaller and harder to hold, and the routes get technically and physically harder.

By the time you get to those harder routes, though, you will have built up the upper and lower body strength that would allow you to climb them. But for the easier routes, and for beginner climbers, you can just start climbing without worrying about having prior upper or lower body strength.