Given your project is linked to a git repo already, you can just use the standard IntelliJ feature to "delete" the file.
In the project pane, focus the file and hit Delete key. The "Safe Delete" popup will appear, click OK.
Now observe under 9: Version Control -> Local Changes that the file is listed there in "grey" - when you commit and push to your git repo, the file will be deleted on the current branch.
Edit: if these are IntelliJ files, this becomes more difficult.
First, close IntelliJ, make a list of the exact files you want to delete from repo, and take a copy of those files on your local file system.
Go to File Menu-> Settings | Version Control | Confirmation, then check the Show options before adding to version control setting under the When files are created section. Alternatively, you could check Do not add. It is mandatory that you do not check Add silently.
Delete the file you don't want to be tracked by VCS.
Press Ctrl + Z to undo the removal of the file. If IDE shows a popup window which lets you choose whether or not to add the new file to VCS, click No.
Commit your local changes, and the files are removed from VCS.