Not necessarily. When we say "I'm going to work", it usually means we're travelling to another place in order to do our job. If she works from home, "I'm working" is the most likely thing for her to say.
I've just had a thought. Did you mean "going to" to express the future? If so, it's grammatically correct but not very natural. The same could be expressed by "I will work" but that wouldn't be natural either.
Don't ask what somebody 'can' say. Ask instead what the best way of saying it is.
Esther's reply is a good example of the present continuous for arrangements. I think what's confused you here is that Robin uses 'going to' in his question.
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