Is it possible to convert output of TV Channel (set top box 720p) to 1080p?
What you are talking about is upscaling and any current HDTV will do upscaling automatically.  Upscaling doesn't work miracles though, it will only make it so that the lower quality signal can be watched on a higher quality display.  It just multiplies the pixels so that a 720 by 480 (.9 pixel compressed) signal for example doesn't end up only taking up 1/4 of your screen.  It doesn't make the video any higher quality (it in fact will be slightly lower quality than if you were to watch it at the native resolution).

Your problem however is not lack of upscaling.  Your problem is using the wrong signal.  You are currently connecting your cable box to your TV through what is known as a composite cable.  It's an old type of analog connection that includes all the video information on the single yellow pin and sends left and right audio on the white and red pins respectively.  It is only capable of reproducing a standard definition signal so you are not actually using HD signals.

To benefit from 720 or 1080 signals, you will have to either use analog Component video (the red/green/blue connectors) or use an HDMI cable.  It's worth noting that as long as your TV and cable box both have component inputs/outputs, you can actually re-purpose the existing Composite cable to carry a component signal, but there will be no audio (you need a separate white/red pair for the audio, which is the same format as with the composite signal).

Once you fix the cable and hookup issue, you should see better image quality.  Personally, I'd recommend using HDMI if it is available on both your cable box and TV as that is the easiest to setup and will give a nice digital signal.