No. It is very improper to try to speak to a judge privately about your case.
In rare circumstances, you may be able to make an application without first serving the other party. This is called an “ex parte” application and there are very limited situations where these are allowed. However, this is still not talking to a judge “privately.” You would need to first file the correct court forms to make an application, and you would speak to the judge in the courtroom, just without the other party present. What you say in court would be recorded.
In every other case, you can only speak to the judge in the courtroom on your scheduled hearing date. This means the other party will know about the hearing and may be there as well.