Endeca comes with a reference application thro which you can easily check the request/response that is to/from the Endeca engine. This is the best way to get the request/response to the extent of my knowledge.
But you can try the following approach (not very sure it this suits your app).
In the ATG application that is integrated with Endeca, you can display the ENEQuery that is built with your search term.
In the implementation class of the com.endeca.soleng.urlformatter.QueryBuilder interface, in which you build the endeca query, you can use logDebug statement to get the query that is generated in the console (make sure that this class extends GenericService).
After building this query you would be calling getQueryResults methods in the Tools class for getting the results which returns ENEContentResults. You can get ENEQueryresults from ENEContentResults. In between any of these lines of code you can choose to display the result.
Hope this helps.
Keep posting your updates/ queries.
Edited by: Gopinath Ramasamy on Dec 5, 2012 7:26 PM
NOTE: The above info is given with reference to an ATG app which is externally integrated with Endeca. With 10.1.1 ATG comes with Endeca search integration OOTB. Please consider this point while reading this post.