It is possible to list a start-up company on NASDAQ or any other stock
exchange to raise the capital for its expansion? If yes, can you
kindly provide me the route to go ahead?

Yes. Certain stockbrokers specalize in raising money.  They take at
least 10 per cent. Private placements are also possible.  What country
and what business are you located in? Some are better then others for
finding money. Are you able to go on a roadshow pitching the company
to stockbrokers who will in turn pitch your stock?
(Some managements are so involved in keeping the company together they
can not take time to sell the company.)

A startup will likely not meet the listing requirements for the NASDAQ
or another exchange.

To be in the public markets, you would like have to use an OTCBB.

Without knowing the details of the startup though, this is really speculation.

Quite so.   Here are the NASDQ listing requirements:
http://www.nasdaq.com/about/nasdaq_listing_req_fees.pdf

And here, those for the OTCBB:
http://www.venturelawcorp.com/listing_requirements_OTCBB.htm

Much more informal, but the company must already have 40 shareholders.