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This is possible, but you will have to open up some ports on your local computer. Please look into port fowarding if you are into this. You are even able to post your whole website on your computer.
Personally I wouldn’t recommend doing this, because you computer could become more vulnerable and you have to rely on your own home network and computer speed. Scaling is not easy with this setup.
However, there is (some kind of) an alternative. Django uses SQLite as the default database. You are able to use SQLite on Heroku. Please note that SQLite is not build for websites that interact a lot with your database and you cannot push a new build as easily as you would normally do. You will have to export the database first and then rebuild it later as Heroku creates a fresh website each time it builds (and the new database entries will be gone with that).
In other words: I would recommend to just pay for an upgrade of the database or move to another hosting company. I am sorry.