Additional information based on comments:
Some comments have clarified that the production directory cannot be deleted as I originally suggested. Furthermore, the code in the production directory is identical to the code in the development directory. This suggests creating a git repo directly in the production directory and then cloning it as the development folder.
On the production machine:
$ cd <production directory>
$ git init
$ git add .
$ git commit
On the development machine:
$ cd <parent of development directory>
$ rm -rf <development directory>
$ git clone <URL to the production directory>
Note I am assuming that the current development repo does not contain any history which you want to keep. If it does, you can instead add the new production repo as a remote and merge the histories.
On the development machine:
$ cd <development directory>
$ git remote add prod <URL to production directory>
$ git checkout master
$ git pull prod master
Original answer:
You should delete the production directory and recreate it as a git repo. Then connect your development repo the production repo.
One way to do this is simply clone as you have done:
$ cd <parent of production directory>
$ rm -rf <production directory>
$ git clone <development directory or URL> <production directory>
This might be difficult if the development directory and production directory are on different machines, depending on the network configuration. Even if you do this, you will need to connect the development directory to the production one in order to push updates:
$ git remote add prod <production directory or URL>
Since you need to do this anyway, you can do it from the beginning. On the production server. First create an empty production repo:
$ cd <parent of production directory>
$ rm -rf <production directory>
$ mkdir <production directory>
$ cd <production directory>
$ git init
Now push the development code to production:
$ cd <development directory>
$ git remote add prod <production directory or URL>
$ git push prod master
Note that the location of the production code is mostly irrelevant. You configure a so-called "remote" with this location. This means that the directories do not need to have the same name.