Did the Italian fascists use the S.P.Q.R. motto?

Obviously not. Fascism is right-wing, a kind of authoritarianism, conserving sovereign power to a singular authoritarian or dictator, like Mussolini. SPQR is left-wing, a democratic motto meaning "Senatus Populusque Romanus" or "The Senate and the Roman People," making it liberal, liberating sovereign power to a plurality, to the people and their body of representatives called a senate, like the Roman Senate, not to a singular authority, like Mussolini.

It was Julius Caesar's overthrow of the Senate and countermanding of the SPQR motto that got him stabbed to death by the Senate after making himself dictator and then emperor, the Senate doing so in order to preserve the republic of Rome and the democratic self-sovereignty of the Roman people. Unfortunately, the Senate assumed that Julius Caesar's nephew Octavian, who had conspired with them to kill him, agreed with them when, in fact, he just wanted to get his uncle out of the way to seize power himself. Octavian changed his name to Augustus Caesar, and having been part of the overthrow of his uncle, knew who his political enemies were, who were they who would most vehemently oppose him becoming emperor in order to preserve the Senate and self-sovereignty of Roman citizens, and had them all killed, thus installing himself as Emperor of Rome and ending the Roman Republic, whose moto was SPQR, forever. The Senate still continued to exist in the Roman Empire, but it had no real power, though it did still hope for its return and still uphole the motto SPQR, it no longer being a declaration of the state of things but merely a stated wish to return to how things were before.