How to write Junit test cases for EntityManger createNativeQuery with String and class as parameters

According to the documentation, if you are using argument matchers, all arguments have to be provided by matchers. Hence, try using ArgumentMatchers.eq() (reference documentation) to check for Designator.class equality:

@Mock
TypedQuery<Designator> getQuery;

@Test
public void saveDesignatorTest() throws Exception {
    Mockito.when(entityManagerMock.createNamedQuery(ArgumentMatchers.anyString(), ArgumentMatchers.eq(Designator.class))).thenReturn(getQuery);
    Mockito.when(getQuery.getResultList()).thenReturn(designatorEntityList);
    assertNotNull(designatorDaoImpl.getDesignators(dto));
}

Avoid using property injection and use constructor injection instead:

@Repository
public class DesignatorDaoImpl extends GenericDaoImpl implements DesignatorDao {

    private EntityManager em;

    public DesignatorDaoImpl(EntityManager em) {
        super();
        this.em = em;
    }

    public List<Designator> getDesignators(DesignatorRequestDTO dto) throws Exception {
        List<Designator> designatorsList = new ArrayList<>();
        int start = dto.getStart();
        String queryString ="select * from ( SELECT * FROM TAB.DESIGNATOR WHERE ACTIVE_IND ='A' ORDER BY LAST_EDITED_DATE DESC ) OFFSET "+ start +" ROWS FETCH NEXT "+dto.getLimit()+" ROWS ONLY";          
        Query query = em.createNativeQuery(queryString,Designator.class);
        designatorsList = query.getResultList();
        return designatorsList;
    }
}

Then you need to drop @InjectMocks because you are injecting them before being initialized and create the instance of DesignatorDaoImpl to test in the @Before method:

@Transactional
@Rollback(true)
public class DesignatorDaoImplTest {

    @Mock
    EntityManager entityManagerMock;

    @Mock
    TypedQuery<Designator> getQuery;

    DesignatorDaoImpl designatorDaoImpl;

    DesignatorRequestDTO dto = new DesignatorRequestDTO();
    List<Designator> designatorEntityList = new ArrayList<>();

    @Before
    public void setUp() throws Exception { 
        MockitoAnnotations.initMocks(this);
        
        dto.setLimit(4);
        dto.setStart(1);

        Designator entity = new Designator();
        entity.setId("dgbkk");
        designatorEntityList.add(entity);

        designatorDaoImpl = new DesignatorDaoImpl(entityManagerMock);
    }

    @Test
    public void saveDesignatorTest() throws Exception {
        Mockito.when(entityManagerMock.createNamedQuery(ArgumentMatchers.anyString(), ArgumentMatchers.eq(Designator.class))).thenReturn(getQuery);
        Mockito.when(getQuery.getResultList()).thenReturn(designatorEntityList);
        assertNotNull(designatorDaoImpl.saveDesignators(dto));
    }
}

If you can't change DesignatorDaoImpl then use @RunWith(MockitoJUnitRunner.class) instead of MockitoAnnotations.initMocks(this);as follows:

@Transactional
@Rollback(true)
@RunWith(MockitoJUnitRunner.class)
public class DesignatorDaoImplTest {

    @Mock
    EntityManager entityManagerMock;

    @Mock
    TypedQuery<Designator> getQuery;

    @InjectMocks
    DesignatorDaoImpl designatorDaoImpl;

    DesignatorRequestDTO dto = new DesignatorRequestDTO();

    List<Designator> designatorEntityList = new ArrayList<>();

    @Before
    public void setUp() throws Exception { 
        MockitoAnnotations.initMocks(this);
        dto.setLimit(4);
        dto.setStart(1);

        Designator entity = new Designator();
        entity.setId("dgbkk");
        designatorEntityList.add(entity);
    }

    @Test
    public void saveDesignatorTest() throws Exception {
        Mockito.when(entityManagerMock.createNamedQuery(ArgumentMatchers.anyString(), ArgumentMatchers.eq(Designator.class))).thenReturn(getQuery);
        Mockito.when(getQuery.getResultList()).thenReturn(designatorEntityList);
        assertNotNull(designatorDaoImpl.saveDesignators(dto));
    }
}