Can I paint the background of my gas fireplace?
Yes.  These are painted all the time.  I just used this for a traditional wood burning fireplace.  You can search for fireplace paint but really you are looking for it to withstand the max temperature.  Since yours is gas I am guessing your max is well less than 1000 F.  

Also most fireboxes are painted black so you can see the fire easier.  And because of the heat a lighter color may get burn marks in it from the heat.