Consider marking one of the beans as @Primary

The correct approach:

public interface SortAlgorithem {
    public int[] sort(int[] arrayNumbers);
}

@Component("Bubble")
public class BubbleSort implements SortAlgorithem {
    Log log = LogFactory.getLog(BubbleSort.class);

    public int[] sort(int[] numbers) {
        log.info("Bubble sort is called");
        return numbers;
   }
}

@Primary
@Component("Quick")
public class QuickSort implements SortAlgorithem {
    Log log = LogFactory.getLog(QuickSort.class);

    public int[] sort(int[] numbers) {
        log.info("Quick Sort is called");
        return numbers;
    }
}

and then you need to use your implementations like this:

@Autowired
@Qualifier(value = "Bubble")
private SortAlgorithem bubbleSort;

@Autowired
@Qualifier(value = "Quick")
private SortAlgorithem quickSort;

You have annotated a field with @Autowired and @Qualifier, but you have also created a constructor which sets the field.

I think that Spring is using the constructor, but doesn't automatically know that the constructor parameter corresponds to the annotated field.

So move the annotations into the constructor declaration:

private SortAlgorithm sorter;

@Autowired 
public BinarySearchImpl(@Qualifier("quick") SortAlgorithm sorter) {
     this.sorter = sorter;
}

Alternatively, you could use a zero-arg constructor, keep your field annotation and let Spring inject using reflection. However in my opinion constructor-injection is better -- it allows you to unit test cleanly, without involving Spring or reflection.

As other answers point out, there are other ways to disambiguate autowired beans -- and the Spring docs explain them all -- but using qualifiers like this does work.