No, Quebec is a province of Canada just as Gujarat or Punjab is the state of India. But under an agreement with Canada, Quebec runs its own immigration program.
Why should I immigrate to Quebec instead of Canada?
Permanent immigration processing time of Quebec is far less than that of Canada. Where, the processing time for permanent residency to Canada takes 3-4 years, it is only 18-24 months for Quebec. Moreover, it is not necessary that you have to stay in Quebec after getting permanent residency. You can move anywhere in Canada and earn your livelihood.
Then why all persons do not apply for Quebec instead of Canada? 

First, eligibility criteria of Quebec are higher than that of Canada. Second, you have to learn French as the interview is conducted in the language.

What does permanent resident status of Quebec offers?

The holder of a permanent resident visa and his/her accompanying dependants are permitted to permanently reside any where in Canada, even outside Quebec and earn a livelihood. They may attend primary and secondary education institutions in the various provincially administered public school systems, tuition exempt up to secondary level. They also qualify for provincially administered universal health care coverage.

Who qualifies for permanent residence under the Economic class?
(Quebec skilled worker class, entrepreneur class, investor class, self-employed persons class)
Under the Quebec skilled worker class, applicants may become permanent residents on the basis of their proven ability to become economically established in Quebec after being assessed under 9 factors.
The Investor class is point based and confers permanent residence to applicants who demonstrate an ability to become economically established in Quebec on the basis of their business or management experience and high personal net worth. Approval is contingent upon the investor undertaking to commit an irrevocable, passive, non-interest bearing investment of $400,000 (Rs.1.4 crore) in a government guaranteed investment fund.
The Entrepreneur class is point based and confers permanent residence to applicants who demonstrate an ability to become economically established in Canada on the basis of their business experience and personal net worth. Approval is contingent upon the entrepreneur undertaking to invest and become active in the management of a qualifying business operated in Canada that will contribute to the economy and create employment.
The Self-Employed class refers to applicants who have the intention and the ability to create their own employment and make a significant contribution to the cultural, artistic or athletic life of Canada, or to create their own employment by purchasing and managing a farm in Canada.
How are applications assessed under the skilled worker class? 
Applications are assessed under 9 Factors (education, employment status in Quebec, experience, suitability, languages known, age, children, spouse characteristics and finance). In order to qualify the unmarried and the married applicant have to score 50 and 58 respectively without counting the points