Will remote URL for fetch and push be different?

Yes (using different remote), and that is why Git 2.5 introduces a new ref shorthand @{push}.
See “Viewing Unpushed Git Commits”

What commands can you use to change remote URL for fetch or push separately?

You need a separate remote:

git remote add myfork /url/for/my/fork
git config remote.pushdefault myfork

The GitHub blog post “Improved support for triangular workflows” illustrates the use of @{push}:

See what commits you’ve added to your current branch since the last push:

git clone https://github.com/YOUR-USERNAME/atom
cd atom
git config remote.pushdefault origin
git config push.default current

• remote.pushdefault specifies where to push (to which remote repo).
• push.default specifies what to push (what refspec), when no refspec is explicitly given.
  current, in that latter case, means “push the current branch to update a branch with the same name on the receiving end.”

The following branch will fetch from one url, push to another:

git remote add upstream https://github.com/atom/atom
git fetch upstream
git checkout -b whizbang upstream/master

(Here the whizbang branches tracks upstream/master, but pushes to origin/whizbang)

git log @{push}..

This uses the new @{push} notation, which denotes the current value of the remote-tracking branch that the current branch would be pushed to by git push, namely origin/whizbang.
You can also refer to the push destination of an arbitrary branch using the notation whizbang@{push}.