Yes as far as immigration is concerned: an L-2 spouse is employment-authorized incident to status. State cosmetology licensing is a separate professional requirement she must meet before working.
Immigration permission and professional licensing are two different gates
The spouse of an L-1B transferee is admitted in L-2 status and, under current USCIS policy, is employment-authorized incident to that status without needing to apply for a separate work permit. She can therefore accept employment at a salon or work for herself for as long as both spouses hold valid status. What immigration does not do is license her trade: hairdressing is regulated by state boards, and most states require their own licence or a recognised equivalent before she can cut hair for pay.
She should ask the receiving state's board about reciprocity for a Quebec qualification and start that process early, because it may take longer than the L-2 admission. Children admitted as L-2 dependants may attend school but are not employment-authorized. If the assignment is seasonal and the family returns each year, her U.S.
work ends when the L status ends.