How long it will take for your hair to turn completely gray (if it ever does) is hard to say.
This process can occur gradually over many years - even decades - as individual hair follicles stop producing color, or it can happen within a matter of months. I've seen people who are completely gray by the age of 20, and others who still have dark hair well into their 70's and 80's. If you have a close relative who went gray at an early age, you may too, although how strong that genetic link is needs to be studied further.

Hairs don't actually "turn" gray - they grow in that way. Every day hairs fall out and new ones emerge in their place. As the hair grows in the follicle, color is deposited into the new growth in the form of two substances, melanin and pheomelanin. Melanin produces the hair shades blond, brown and black, depending on the concentration of pigment in your hair. Pheomelanin produces red hair and the reddish undertones seen in blond, brown and black hair. When your body stops producing these colored pigments, the hair growing out of the follicle will grow in gray.