The phrase "a note to the judge" is correct and usable in written English.
You can use it when referring to a written communication intended for a judge, often in a legal context.
Example: "The lawyer submitted a note to the judge requesting a reconsideration of the bail amount."
Alternatives: "a message for the judge" or "a letter to the judge."
A juror sent a note to the judge on Monday asking to be excused.
In a note to the judge, the jurors in the Fortunato case said tensions were rising.
The jurors had rough moments, but, after one sent a note to the judge, they kept talking.
Within hours, jurors asked to go home, sending a note to the judge saying things had become "stressful".
But the next day, shortly after the jurors had begun to deliberate, they sent a note to the judge.
In the end, Mr. Matsumoto sent a note to the judge, his fifth over 14 days: A verdict was in, such as it was.
Mr. Davis's mother, Mary Davis, who attended every day of the 10-week trial, left the courtroom in tears, shouting, "Yes, Lord! Yes, Lord! Yes, Lord! Praise Jesus!" An Earlier Deadlock Earlier yesterday the jurors sent a note to the trial judge, Justice Bernard J. Fried, saying they were deadlocked.
In a note to the presiding judge, the jurors said they "cannot arrive at a unanimous decision concerning the defendants, Pedro Espada Jr. and Pedro G. Espada".
The jury considering allegations against a reputed Mafia soldier on charges of conspiring to kill Salvatore Gravano, the Mafia turncoat, sent a note to the trial judge late yesterday saying that it was deadlocked.
The jury sent a four-part note to the judge requesting clarification on what it would take to find the defendant, Lemrick Nelson Jr., guilty of "aiding and abetting" the violation of Yankel Rosenbaum's federal civil rights.
After lunch as reporters and spectators swarmed back into the courtroom in anticipation of a mistrial, the jury sent another note to the judge.
