Both are grammatically correct, but - if you think about it - only the first one is appropriate for the situation.
1) Are we getting close to London?
This is a normal question to ask. It means "Are we near London now?"
2) Are we getting closer to London?
Taking the meaning literally, this would be a strange question to ask - because the answer is evidently 'Yes'. If the train is travelling in a south-easterly direction from Manchester to London, every inch that you travel is obviously taking you closer to London.