Perhaps. There are two main areas of a bicycle prone to damage in truly cold temperatures.
The first is your freehub. When the grease in a freehub gets thicker, it can prevent the hub from full engagement. At best this means when you pedal forward, nothing happens. At worst, you can get partial engagement and chip/destroy the teeth inside your freehub, destroying it. Winterize your freehub or risk this type of damage.
The second is suspension. Most suspension is designed to dissipate heat and doesn't deal particularly well with the cold. Cheap elastomer suspension will often crack and disintegrate, leaving a springing non dampening system behind. Oil and air systems often have seals that can deform in the cold or crack, rendering the suspension non-functional for either the short term or permanently.
Most plastics, metals, and carbon fiber will work just fine down to -40C (-40F), which is colder than most people are willing to ride. No part on you bike is going to magically shatter because it gets too cold. Any lubricated part (your chain, hubs, steering, pedals, bottom bracket, derailleurs ) will work, but may work less well because the lubricants get stiff (or solid). Except for your freehub/freewheel (which can be completely destroyed) none of these other areas are likely to sustain damage. The lubricants will simply slow/freeze them and they won't work until they are warmed up.
Most suspension (depending on type) will stop working at some temperature. Some leaf sprung style systems will not, but most systems will freeze. Depending on the quality/specs of other parts (seals, tolerances, etc) some systems may be damaged by continuing to use them after they freeze, others will survive to work again when temperatures rise.