No. TD status allows a spouse or unmarried child under 21 to accompany the TN professional and to study, but it does not authorize employment. A spouse who intends to work needs a route of their own.
What usually goes wrong
The common error is treating a Québec employer's willingness to keep the spouse on remote payroll as permission. Employment is judged by where the person is physically performing the work, not by where the employer is registered, so a TD spouse logging in from a U.S. apartment is working in the United States without authorization.
Before the move, the spouse should either identify an independent route, for instance a listed profession of their own under TN, or plan not to work while in TD status. Children in TD may enrol in school, and that is the one activity TD clearly permits beyond residing with the principal. Record each family member's citizenship, because a spouse who holds Canadian permanent residence rather than citizenship will follow a different derivative process at a consulate.
Do not let the engineer's start date fix the spouse's plan; the spouse's own status question deserves its own answer.