Option A:
git clone git@github.com:whatever folder-name
Ergo, for right here use:
git clone git@github.com:whatever .
Option B:
Move the .git folder, too. Note that the .git folder is hidden in most graphical file explorers, so be sure to show hidden files.
mv /where/it/is/right/now/* /where/I/want/it/
mv /where/it/is/right/now/.* /where/I/want/it/
The first line grabs all normal files, the second line grabs dot-files. It is also possibe to do it in one line by enabling dotglob (i.e. shopt -s dotglob) but that is probably a bad solution if you are asking the question this answer answers.
Better yet:
Keep your working copy somewhere else, and create a symbolic link. Like this:
ln -s /where/it/is/right/now /the/path/I/want/to/use
For your case this would be something like:
ln -sfn /opt/projectA/prod/public /httpdocs/public
Which easily could be changed to test if you wanted it, i.e.:
ln -sfn /opt/projectA/test/public /httpdocs/public
without moving files around. Added -fn in case someone is copying these lines (-f is force, -n avoid some often unwanted interactions with already and non-existing links).
If you just want it to work, use Option A, if someone else is going to look at what you have done, use Option C.