Franco was from Ferrol, in Galicia, and I'm not sure if he spoke Galician as well as Spanish. However, I do know that he suppressed other languages in Spain that were not Spanish such as Basque and Catalán. Did he do the same with Galician?(Galego)
For a really short, unsatisfactory answer: yes, the Galician language was also repressed, and arguably more successfully than in Catalonia or the Basque Country. The continued use of Galician was never institutionalised or part of elite culture in the same way it became in Catalonia in particular. Even by the end of the Franco regime, there was a conscious rejection of Galician literary heritage on the part of local officials - Andrew Dowling quotes the mayor of Galicia's capital Santiago de Compostela as dismissing Castelao (the region's most famous Galician-language writer) as being 'barely deserving of a public lecture' in 1975. An equivalent statement in a Catalonian context would be just about unthinkable by that point.
Dowling points to two factors to explain this divergence: the hostility of local clergy towards the Galician language (in contrast to Catalonia, where the insistence on continuing to hold mass in Catalan in the early period of the Franco dictatorship helped keep the language in popular and official use), as well as the relative historical weakness of Galician nationalism. I'm not really in a position to evaluate either claim - Dowling is an expert on Catalonia, and his claims about Galicia are made in passing rather than as the focus of his investigations. I'm personally even further from being an expert in the regional politics of Galicia, and most of the specialist literature I'm aware of is inaccessible to me for reasons of location and language (I don't speak Galician...).
The source I've referred to is from Andrew Dowling, 'Prohibition, Tolerance, Co-option: Cultural Appropriation and Francoism in Catalonia, 1939–75', Contemporary European History 27:3 (2018), pp. 370-86.