No. A family residence permit does not by itself confer the right to work; a separate work permit is required. The family residence permit under the Law on Foreigners and International Protection (No. 6458) gives your spouse the right to remain lawfully in Türkiye, but employment requires a work permit under the International Labour Force Law (No. 6735).
That said, a family residence permit is a legitimate basis of stay that makes the work permit application easier, and for those who have held one for a certain period the application conditions can be comparatively favourable. Check the validity and the remaining term of the family residence permit first, then run the work permit process separately through the intended job and employer. Making the employment contract conditional on the permit being granted, and not starting actual work before it is issued, keeps administrative penalties off the table.
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