To pick up the tab means to pay the bill, normally for yourself and at least one other person. You usually pick up the tab for a meal or for some drinks in a bar for you and a friend or a group of friends. It's an act of generosity on your part.
If you go to the gym or to a swimming pool with a friend and you are both expected to pay an admission fee, and you offer to pay for both of you, you could describe it as picking up the tab but it would be an unusual use of the phrase.
As bhaisahab said, it sounds like you are offering to pay the bills for the gym! That would be quite expensive. I imagine a gym spends a lot of money every month on water, electricity, lighting, heating, equipment maintenance ...!