Yes, you can marry your girlfriend in Canada as a traveler and she as a visa student; however, the marriage will not change your visa status for both you and your girlfriend.
You have to marry to a Canadian citizen or a permanent resident, and the person that you're marrying with should, by law, have financial prove, either a source of income or assets, that your spouse has the financial capability to support your stay during the immigration process; which, can take up from one to three years, depending on your situation.
If you want to get more information on this topic, you can visit http://www.cic.gc.ca/english/immigrate/sponsor/spouse.asp
Hope it helps. Good luck.