how can we query foreign key and get results of objects which are connected to foreign key

You can combine both the title_count and category_count methods under one Manager. When filtering through a ForeignKey, your field lookup needs to specify the name of the field you're trying to filter on, else it will assume you're querying the ID field. So instead of category__icontains, you would do category__title__icontains.

Another note, in newer versions of Django, you are required to specify the on_delete parameter when defining a ForeignKey.

This is a working example of what I think you're trying to accomplish.

class BookManager(models.Manager):
    def title_count(self, keyword):
        return self.filter(title__icontains=keyword).count()
    def category_count(self, keyword):
        return self.filter(category__title__icontains=keyword).count()

class Category(models.Model):
    title = models.CharField(max_length=20)
    def __str__(self):
        return self.title

class Enquiry(models.Model):
    title = models.CharField(max_length=200)
    category = models.ForeignKey(Category,
                                 default=False,
                                 blank=False,
                                 on_delete=models.CASCADE)
    detail = models.TextField()
    books = BookManager()
    # tags=models.ChoiceField()
    def __str__(self):
        return self.title

Here's how you would use it:

Enquiry.books.category_count('python')
Enquiry.books.title_count('test')