What is the correct punctuation surrounding the quoted question in this example:
When a neighbor asks, "When are we getting together?" that is the beginning of building community.
Your example is correct in both American and British English.
In English:
Therefore the following are all correct:
When a neighbor says, "We are getting together," that is the beginning of building community. (American English)
When a neighbor says, "We are getting together", that is the beginning of building community. (British English)
When a neighbor asks, "When are we getting together?" that is the beginning of building community. (British and American English)
But it looks strange not to have commas on both sides of the quotation, and in fact the last clause has a longer pause before it, when you speak it, and might be better offset with a dash:
Getting together – that is the beginning of building community.
When a neighbor asks, "When are we getting together?" – that is the beginning of building community.