I have to wear a hear monitor for a month,can i still workout at the gym?

Yes: stop if: uncomfortable or short of breath/chest pain.

Certainly: It is a good idea for you to wear it through every one of your normal activities. It is there to catch any variation in rhythm & these can be triggered by the varied natures of stresses induced by the activities. If you don't wear it for all your common activities it may miss something important.