How do I find the number of child terms associated with a specific GO term?
I don't think it's possible to do this via QuickGO. You can, however, do it from GO's website, using their GOOSE tool. This offers an interface to the GO SQL database and lets you run queries. It also offers example searches, one of which ("Find descendants of the node 'nucleus'") can easily be modified to return what you are asking for:

SELECT COUNT(*)
FROM
 term
 INNER JOIN graph_path ON (term.id=graph_path.term1_id)
 INNER JOIN term AS descendant ON (descendant.id=graph_path.term2_id)
WHERE term.name='heart development' AND distance = 1 ;


In the query above, I modified the example and changed 'nucleus' to 'heart development' (GO:0007507) and changed distance &lt;&gt; 0 to distance = 1 to make it list only direct descendants. Finally, I also changed SELECT DISTINCT descendant.acc, descendant.name, descendant.term_type to simply SELECT COUNT(*) since all we are interested in is the number of descendants. If you also want their names, leave the SELECT line as is. That will show both the list of names and the total in the output. 

The modified query shows us that this term has 30 descendants: