How can Mockito capture arguments passed to an injected mock object's methods?

Use one, or more, ArgumentCaptors.

It is unclear what your types are here, but anyway. Let's suppose you have a mock which has a method doSomething() taking a Foo as an argument, then you do this:

final ArgumentCaptor<Foo> captor = ArgumentCaptor.forClass(Foo.class);

verify(mock).doSomething(captor.capture());

final Foo argument = captor.getValue();

// Test the argument

Also, it looks like your method returns void and you don't want it to do anything. Just write this:

doNothing().when(theMock).doSomething(any());

You can hook doAnswer() to the stub of the send() method on amqpTemplateMock and then capture the invocation arguments of AmqpTemplate.send().

Make the first line of your testDoSomething() be this

    Mockito.doAnswer(new Answer<Void>() {
          @Override
          public Void answer(final InvocationOnMock invocation) {
            final Object[] args = invocation.getArguments();
            System.out.println("UUID=" + args[0]);  // do your assertions here
            return null;
          }
    }).when(amqpTemplateMock).send(Matchers.anyString(), Matchers.anyObject());

putting it all together, the test becomes

import org.junit.Before;
import org.junit.Test;
import org.mockito.InjectMocks;
import org.mockito.Matchers;
import org.mockito.Mock;
import org.mockito.Mockito;
import org.mockito.MockitoAnnotations;
import org.mockito.invocation.InvocationOnMock;
import org.mockito.stubbing.Answer;

public class UserServiceTest {

  /** This is the class whose real code I want to test */
  @InjectMocks
  private UserService userService;

  /** This is a dependency of the real class, that I wish to override with a mock */
  @Mock
  private AmqpTemplate amqpTemplateMock;

  @Before
  public void initMocks() {
    MockitoAnnotations.initMocks(this);
  }

  @Test
  public void testDoSomething() throws Exception {
    Mockito.doAnswer(new Answer<Void>() {
      @Override
      public Void answer(final InvocationOnMock invocation) {
        final Object[] args = invocation.getArguments();
        System.out.println("UUID=" + args[0]);  // do your assertions here
        return null;
      }
    }).when(amqpTemplateMock).send(Matchers.anyString(), Matchers.anyObject());
    userService.doSomething(Long.toString(System.currentTimeMillis()));
  }
}

This gives output

UUID=8e276a73-12fa-4a7e-a7cc-488d1ce0291f

I found this by reading this post, How to make mock to void methods with mockito