I did not realize Triluma needed to be refrigerated, can I continue to use it if it was not?

Yes you can, stability of HQ is not compromised if it has been left out of the fridge for days to weeks. The active components are still viable, so yes it you can continue to use it. Melasma can not be completely cured, however effective treatment are possible. This is because your skin is extremely sensitive to UV and even the smallest amount can stimulate your pigment cells to produce colour. The mainstay of treatment is strict UV protection- hats, sunglasses and SPF every 4 hours. 
I prefer to use a combination of creams and laser for melasma. Hydroquinone can be used (5-8%), along with vitamin A creams. I usually start my patients on laser (low dose Q switch or Picosure) a few weeks after they commence on creams. Another laser I use is the Clear and Brilliant in the Permea setting combined with creams. 
In some cases I combine glycolic AHA peels, and in other cases I use a tablet called Tranexamic acid to help. Melasma treatments should be tailored to each patient. 
All the best, Dr Davin Lim. Laser, surgical and aesthetic dermatologist. Brisbane, Australia