I think the question, as worded, has some incorrect assumptions built into it, but let me try to hit the key answers that I think might help:
Your broker can't really do anything here. Your broker doesn't own the calls you sold, and can't elect to exercise someone else's calls.
Your broker can take action to liquidate positions when you are in margin calls, but the scenario you describe wouldn't generate them: If you are long stock, and short calls, the calls are covered, and have no margin requirement. The stock is the only collateral you need, and you can have the position on in a cash (non-margin) account. So, assuming you haven't bought other things on margin that have gone south and are generating calls, your broker has no right to do anything to you.
If you're wondering about the "other guy", meaning the person who is long the calls that you are short, they are the one who can impact you, by exercising their right to buy the stock from you. In that scenario, you make $21, your maximum possible return (since you bought the stock at $100, collected $1 premium, and sold it for $120. But they usually won't do that before expiration, and they pretty definitely won't here. The reason they usually won't is that most options trade above their intrinsic value (the amount that they're in the money).
In your example, the options aren't in the money at all. The stock is trading at 120, and the option gives the owner the right to buy at 120.* Put another way, exercising the option lets the owner buy the stock for the exact same price anyone with no options can in the market. So, if the call has any value whatsoever, exercising it is irrational; the owner would be better off selling the call and buying the stock in the market.