Can my wife go rock climbing she's in the first trimester of pregnancy?

You think so??? Yes she can, and no she should not. How will either of you feel if she falls and has a miscarriage? Even if the miscarriage has nothing to do with the fall, you will always wonder if she would have had a miscarriage if she did not fall. Enjoy her pregnancy...There will be plenty of time for rock climbing later.
No. Any potentially dangerous exercise/activity is contraindicated in pregnancy. Pregnant women have poor coordination of their movements as a result of pregnancy hormones and are more prone to falls and injury than non-regnant women. Avoid rock-climbing, all contact sports and any strenuous exercise or heavy lifting in pregnancy! try swimming, stationary bike, brisk walking/treadmill in a/c spaces.