At what age does a filly become a mare?

Five years old.

A filly is a young female horse too young to be called a mare. There are several specific definitions in use.

In most cases filly is a female horse under the age of four years old. In some nations, such as the United Kingdom, the world of horse racing sets the cutoff age for fillies as younger than five years.
https://en.wikipedia.org/wiki/Filly

[Added reference - McG]

Four years old.

At what age is a filly recognized as a mare? A filly is usually determined between the ages of 1 year and under 4 years; while a mare is 4 years or older. A filly can be physically able to become pregnant at the age of 2, but it is highly recommended to wait until she has stopped growing herself, which is between the ages of 3-4 years.
http://www.equineworld.co.uk/about-horses/horse-gender-definitions

[Added reference - McG]