How to connect with a professor?
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.