I am using internet behind NAT, of my institute. I have no access to router's username and password. I want to host a website from my laptop. Is there any way I could host it?
That really depends on the configuration of the main router, but the answer is probably "no". Most likely main router doesn't forward any ports to your laptop.
Of course there are workaround, like for example, setting a VPN or SSH tunnel connection to another server that will forward ports to your computer, but if you'll have access to the server that is already accessible from the internet - you'll just host your site there.