I am interested in learning differential topology as Milnor, Guillemin, Pollack, Hirsch, Kosinski, etc.. did it. However, I am in University of Toronto as an undergraduate and none of colleges (or universities in Canadian English) near me do not have any professors working in that field. Is it the same trend in America? Or did a lot of things happen since 1960-70s (back then that area was flourished)?
What you say isn't really true.
There's plenty of people at U.Toronto that understand that material. V. Kapovich is one. But plenty of people in symplectic geometry/topology and the dynamics group know that material as well. I'm at a Canadian university (U.Victoria) and I know that material quite well. A bunch of people at McMaster know that material well, also.
As a field, Differential Topology mostly began in the States, France and Russia. Germany, Switzerland, Denmark, the UK, Japan, etc, all these countries have plenty of people that know or developed that thread of ideas to some extent. But the bulk of that field is in the States. Not many people call themselves differential topologists anymore, mostly because as a field it's evolved into more specialized niches, all with their own names.
Also, "Differential Topology" as a term didn't really catch on in some countries, like Russia. You'll find some field specifiers are culture-dependent. More often than not differential topology is lumped-in with differential geometry by Russians. I apologize if I'm painting with a broad brush but your question is fairly broad.