Your example is correct in American English.

In American English:

• commas go inside the quotations marks
• commas replace full stops but not exclamation or quotation marks

Therefore the following are both correct:

When a neighbor says, "We are 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.

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.

Your example is correct in both American and British English.

In English:

• commas replace full stops in quotations, but not exclamation or question marks

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.