Yes, as I previously stated, if there is a problem with the flyback transformer circuit, it is possible that the frequency or volume of the noise it generates can become increased or different.
Though again, PC monitors never made an audible noise unless they were low quality and used the cheaper 15.7kHz transformer in their construction.
Other noises associated with CRTs are the degaussing noise, which only happens once usually after turning on the CRT or after pressing the degauss button, or the sound of old IDE hard disks spinning, which also make a constant high frequency noise.