Many microwaves have a feature that allows you to select which of several sounds or tunes they play as an alert tone. Some of those include the option to use no sound at all. However, that setting is not default, so if there's a power failure it may reset itself to some sound and need to be adjusted again.
For normal operation the major sound is the cooling fan for the magnetron tube, and I don't know of any that can avoid using a fan. So silent is highly unlikely. Your only choice there is to try to find one that is quieter that others. Don't know if the makers spec the sound level in their info. MAYBE you'd find a quieter one among those that are smaller with a lower power output (and hence requiring less fan cooling), but a lot of those are also made cheaply with little attention to niceties like sound levels.