I propose to use a dedicated package for this named django-postgres-fuzzycount [GitHub]. This package provides a manager that will do the fast counting.
You can install the package with pip (for example in your local environment):
$ pip install django-postgres-fuzzycount
Then you can add the FuzzyCountManager to the models where you want to obtain an approximative count:
from django.db import models
from fuzzycount import FuzzyCountManager
class SomeModel(models.Model):
    #  ... (some fields) ...
    objects = models.Manager()
    approx = FuzzyCountManager()
and then you can count approximatively with:
SomeModel.approx.count()
In case you .filter(..), Django will calculate the real number, since the pg_class table only stores an apprixmative number of rows for the entire table, so:
SomeModel.approx.filter(foo=bar).count()
will take more time (depending on indexes, etc.).
You can also "patch" the objects manager directly, but then obtaining the real number of records might be harder:
from django.db import models
from fuzzycount import FuzzyCountManager
class SomeModel(models.Model):
    #  ... (some fields) ...
    objects = FuzzyCountManager()
It is also nice that if you change the backend database to another database, the FuzzyCountManager(..) will act as a normal Manager, so in case you later change the database system, you do not have to rewrite the managers.