Although I don't know the specifics about post graduate qualifications over there, I can tell you what happens here.
At vet school you maust pass as a vet in all types of animals. One of the reasons is that you must, legally be able to treat any animal in an emergency, also you never know what might happen and its not unheard of for people to change careers later on, or to end up specialising in something else that they never dreamed of doing, whilst at school.
After qualifying you get a job and decide what you want to specialise in. Obviously if you want to do horses you need to get a job in a practice that does horses. Then (over here) you can study for a post graduate qualification (called a certificate) which takes a few years. You attend courses and fill in a case book, and sit exams. The once you have a certificate in a particular area, you might want to do a different one, or study in the same area towards the diploma. In the equine world there are several different certificates you can try for, stud medicine, orthopaedics, surgery etc.... or even just a general practice certificate.
I know things are different in Canada and the US but I expect if you go onto the Canadian veterinary peoples website (whoever they are - sorry I don't know) there will be information on there.