How to connect with a professor? I am a mathematics undergraduate student from India and am presently in my final year. Being a bit less talkative, I had not had many conversations with my professors even though I had tried to maintain a good score in my subjects and learn them well. However nowadays,  when I am trying to apply for my master's I am becoming aware of the need for connections with your professors. However I am not sure how to do this and I have no clue whom to talk to. Isn't it awkward if you just go to a professor's office say Hi and try to start up a conversation?
   So can anyone suggest how to do this? Also is one year (or maybe a semester of 4 months as I have to start applying after that) enough to make good connections with professors? 
Thanks in advance.
From my experience, whether you are outgoing or talkative is not that important. Myself and my former PhD supervisor would notice good students based off the academic merit even when they struggled with English as a second language or if they were a bit introverted. It is not necessarily about your ability to become friends but more your ability to produce quality research when you get trained. A good PhD supervisor will be able to recognise that. 

That being said, depending on the culture of course, my approach was to first send an email outlining your interests and request either a brief discussion, or a coffee when they get a moment. 

I don’t think it is awkward at all to knock on a professors door and say hi and tell them your intentions to become a researcher. 

I am not familiar with how it works in India but in Australia, if someone knocked on my door and was interested in doing some research, I’d buy them a coffee and have a chat if I was free. Otherwise, tell them to come back another time.