One month ago, I had breast reduction.I want to know if my the areola will grow back and when will that be? (photo)

It looks like you have lost some pigment in a portion of the areola.  Hopefully some or all of the color will return in the coming months.  Permanent makeup is an effective option if color does not return to your satisfaction.

It appears that you had some blood supply compromise to the arolea which is a known to occur occasionally. It also appears that you are healing well and that the pink area which lacks pigment may regain it based on the small islands of pigment that are noted.  If the pigment doesn't completely return, tattoo of any residual lighter areas is an option. 

From the photo you provided, it looks like a portion of your areola had some circulatory compromise and lost the superficial layer of skin. This is the layer that contains pigment, so the usual brown color is gone and the areola is pink. It also appears that the pigmentation is returning around the edges and in several small islands. This is a good sign. Often, as the deeper layers of the skin recover, the pigmentation returns. As long as there is no open wound, give the area time to recover, and often the pigmentation returns over several months. If an open wound remains, be certain to follow up with your plastic surgeon for supportive care. If after the area is fully healed the pigmentation still does not return completely, tattooing is also an option.

Blood supply to the nipple areola complex is always at risk during a breast reduction, especially in large reductions and if the change in nipple position is significant. It appears you had some superficial sloughing of the areola and as a result lost pigment. Based on your photo it looks like it is on the road to healing. It is hard to predict the final result and how long it will take to heal. It could take weeks to months. I would check with your plastic surgeon and they will be able to address your concerns. Good luck.