Do you need upper body strength to rock climb?

You do need certain levels of upper and lower body strength, but, the amount of strength really depends on the grade you want to climb.

If you are a beginner, then you don't really need any extra strength to start climbing, as the beginner level routes are very easy. They are similar to climbing ladders, just with some curves.

If you want to climb harder grades, then yes you will need upper body strength, and lower body strength also. Though, by the time you get to these harder routes, you will probably build up that strength anyway through your climbing journey to those grades, and if not, you can include some additional training into your schedule.

Just to point this out though, I have quite a bit of upper body strength as I have been lifting weights for many years on and off, and I climb v7/v8 normally. I have a friend who is significant weaker than me, but, he climbs v8+ and is a much better climber than me. The point is that even if you do not have a lot of upper body strength, you can still be a great climber and climb hard routes. The upper body strength helps, but it's really a combination of strength and technique that will make the biggest difference.