No, they don't do that here.
Earning Premium

kco is wrong. There is a way, but you must commit yourself to cc for quite some time. You must play 1000 greeter games. You must learn the rules of the site, and know when they change, so you can inform everyone you know here. It doesn't matter if staff doesn't tell you, you are required to read TOS every day after login. You must also post in Help and Support at least eight times per day, and your responses must be accurate. While staff rarely replies there (see "bugs" thread), your replies will be monitored.
There were other things, but I stopped reading and paid $29 so I can play whenever I want.

oh. Post in help and support at least eight times a day??? Thats too tough.
For how many months should I do that?

Did you give your credit card number? I have a new chess.com id and they offered me 1 week free premium membership but i should give credit card number.

At one time, IM Danny Rensch offered an opportunity to earn premium if you signed up to relay chess moves from events for live games for Chess.com's coverage of the live events.
I'm not exactly sure what the terms were or if it is even still something available, but I do remember reading his announcement about it.
As far as I recall, you needed to commit to work a certain number of events in order to earn and then maintain your membership privilege.
Maybe you could do some searching in the forums here and see if you can find more details.
I hope this helps.
Is there any way of earning premium membership here?
Like in chesscube, they give away crowns as prizes of tourneys which we can redeem for videos or vip memberships.