In Quora's early days Adam D'Angelo and Charlie Cheever wrote so many questions to fill their site with the needed data, See how Quora started.
According to a thread on Quora early employees kept doing the same thing the founders made which is filling the site with questions, See why Quora is better than Yahoo answers.
Adam D'Angelo , Quora's founder, had many good connections because he was the CTO of Facebook before starting Quora. Those Connections helped Adam get funding, media attention and great employees. See how Facebook started.
Adam had good money because he was a Key employee at Facebook. Adam spent a lot of money on Quora. It was reported that he invested $20 million of his own money in one round.
On one Quora thread Adam said that they invited their friends who also invited their friends and so traction happened.
Quora focused on tech questions in early days. This attracted smart Tech geeks who are smart and dedicated. This kind of community helped the site have high quality intellectual answers, See how Wikipedia started.
According to a Quora thread beta invitations were sent to people who are well connected and who have good knowledge. This helped the site build traction.
In one interview Adam said that Quora focused on the topics that can't be easily found on Google such as information on startups. This strategy helped Quora rank high on Google which led to massive Google traffic, See getting better visibility on Google - Matt Cutts.
Google ranks websites based on their authority. Quora started with so many answers about Tech questions. This resulted in many back-links coming to the site and so Google considered it an authoritative site in the tech field.
It's widely believed that the founders paid some money to get some initial traffic.