Yes, most of the LCD's will support CC.
But I think the answer you seek may be a little more complex. But this will depend on your source medium and how you interact with it.
Incompatibility issues with HDTV
Many viewers find that when they switch to an HDTV they are unable to view closed caption (CC) information, even though the broadcaster is sending it and the TV is able to display it. Originally, CC information was included in the picture ("line 21"), but there is no equivalent capability in the HDTV 720p/1080i interconnects between the display and a "source". A "source", in this case, can be a DVD player or an HD tuner (a cable box is an HD tuner). When CC information is encoded in the MPEG-2 data stream, only the device that decodes the MPEG-2 data (a source) has access to the closed caption information; there is no standard for transmitting the CC information to an HD display separately. Thus, if there is CC information, the source device needs to overlay the CC information on the picture prior to transmitting to the display over the interconnect. Many source devices do not have the ability to overlay CC information, or controlling the CC overlay is extremely complicated. For example, the Motorola DCT-5xxx and -6xxx cable set-top boxes have the ability to decode CC information located on the mpg stream and overlay it on the picture, but turning CC on and off requires turning off the unit and going into a special setup menu (it is not on the standard configuration menu and it cannot be controlled using the remote). Historically, DVD players and cable box tuners did not need to do this overlaying, they simply passed this information on to the TV, and they are not mandated to perform this overlaying. Many modern HDTVs can be directly connected to cables, but then they often cannot receive scrambled channels that the user is paying for. Thus, the lack of a standard way of sending CC information between components, along with the lack of a mandate to add this information to a picture, results in CC being unavailable to many hard-of-hearing and deaf users. "HDMI not allowing Closed Captioning?"
Source Link: http://en.wikipedia.org/wiki/Closed_captioning