There’s no exact equivalent to closing a branch in Git, because Git branches are more lightweight than in Mercurial. Their Mercurial equivalent is more bookmarks than branches.
If I understand correctly, closing a branch in Mercurial roughly makes it disappear from the branch list, so you can achieve the same thing by archiving it. A usual practice is to tag its tip as archive, and delete it:
git tag archive/<branchname> <branchname>
git branch -d <branchname>
git checkout master
The branch will be deleted, and can be retrieved later by checking out the tag, and recreating the branch:
git checkout archive/<branchname>
git checkout -b new_branch_name