Not a chance, for a lot of good reasons.
Garibaldi had not a political bone in his body, and his "achievements" in the south of Italy are there to prove it without any possible doubt (he left quite a political mess afterwards, but then he never claimed to be a man who could build a country), his goal was a united Italy (and creating a competing republic in the south would have defeated his dream, notwithstanding the hopes of Mazzini), he was faithful to VEII (and the sentiment was returned) and finally Cavour would never have allowed it.
Cavour was always skeptic about the idea of the expedition to Sicily, but in the end was forced to countenance it by the public opinion and the king. Therefore Garibaldi was allowed to recruit his volunteers and to put them on two steamers; he provided also rifles and shot (but not the modern Enfield purchased by public subscription: Garibaldi received the older rifles used by the Piedmontese army in 1859), and finally his expedition was carefully steered by Cavour (no landfall in Sardinia, for example: the expedition stopped briefly in Tuscany, as directed by Cavour).
In particular, Cavour always kept a personal distance (and a plausible deniability) from Garibaldi's expedition. He had already a lot on his plate (in particular the relations with Louis Napoleon were a bit frayed, and the status of the duchies in Emilia and the Papal Legations in Bologna and Romagna was disputed). However he played the poor hand he had been dealt very well, and once the rotten Bourbon monarchy in southern Italy collapsed, he made the best possible gains: Marche and Umbria were annexed, as well as the south, and Louis Napoleon was kept friendly (and there was even time for setting up a wonderful piece of skilled political theater when the king and Garibaldi met at Teano). Garibaldi never forgave him from stopping the advance on Rome, but then this goes back to what I was saying earlier about the general's poor political skills.