While there is no signal, the PLL on the monitor will peg itself all the way at one extreme of its range. Typically, that's the lowest possible frequency, which is most likely to be audible to you and will seem louder as you're more sensitive to it. Once the signal shows up, it syncs up and runs at that frequency, which is going to be higher and perhaps inaudible to you.
In other words, it's probably ALWAYS making that "whine" or "whistle" signal; you just can't hear it when there's a signal

It's normal, too. What you're hearing is mostly the core of the flyback transformer moving at ~15kHz, which is just barely audible for many humans. Nothing will break from this.
If it's REALLY loud (like unbearably loud), the core of the flyback may be loose. This is sometimes trivial to fix and sometimes essentially impossible to fix. Just live with it, if possible.