You can retrieve info about a Conference by filtering the Conferences list endpoint by the Conference friendly name. For example if your Conference is named "foo" you can do
GET /2010-04-01/Accounts/AC.../Conferences?FriendlyName=foo
to retrieve a conference.
There is an example in Ruby here:
http://www.twilio.com/docs/api/rest/conference#list-get-example-2