What Tiako said about the lack of evidence is true, but we do have enough information to make an educated guess, and the short answer is propbably not. Slavery in the ancient world did not have the same character that it does in the modern world. When stuyding the ancients it’s important to set aside our notion of slavery which is informed mostly by the antebellum south. This form of slavery was a type of caste system that came into being during the colonial era and was motivated in part along pseudo scientific ideas regarding race. If one was born a slave then he or she would remain a slave for their entire life, ditto for their children. There was no hope of manumission. Slavery in the ancient world did not work this way, both with regards to how one becomes a slave and how their life may play out.
There were several possible events that might see one become a slave: conquest, debt, and brigandry to name a few. Regardless of how it happened, anyone looking at the slave would likely call it a case of bad luck (I believe it was Herodotus who actually used that phrase, though I may be confusing him with another). The point is that it could happen to anyone. Prior to the ascendency of Athens in the classical era for example, slavery was a problem run so rampant that it stunted their economic growth. It reached the point where even aristocrats were selling off their children or being sold themselves because they had incurred too much debt. Much later, the Romans gained the bulk of their slaves through simple conquest, but this practice was normal in the ancient world (albeit the Romans kicked it up to an unparralled degree).
Whatever the reasons someone might become a slave, the important point to note is that they were, one, very common and thought of in the way that we might think of cars, and two, likely to be freed if they were found to be good, loyal workers. There wasn’t (generally speaking) the modern notion of racial superiority and the idea that “a slave is a slave because they’re slave-like.” In fact, if we look at the position that the Greeks held within the collective Roman psyche, one could say the Romans felt positively inferior to their Greek slaves.
If we take all of that and think of Spartacus specifically, it seems unlikely that he would have any type of kinship with fellow slaves simply because they were slaves in the same way that a prisoner in jail wouldn’t really think too much about the guy in the next cell over. But conceptual thinking aside, his actions in italy would seem to indicate that he wanted nothing more than to get back home to Thrace. While the path that he carved through the peninsula has been puzzling scholars for a long time, it doesn’t seem motivated by a campaign of liberty. I believe the simplest explanation is probably the correct one. he didn’t know the terrain all that well and didn’t know how best to escape. First he thought he might head north on foot, then decided his odds might be better going by sea to the south. I personally doubt very much that he ever gave serious thought to overthrowing an institution that was so completely common throughout the history of the mediterranean up to that point.
The problem is we really don't have much information on Spartacus. There is very little in the way of discussion of motivation in what few historical sources we have, because all that information is focused on his Roman antagonists. He is more of a situation than a person. Pretty much the closest thing we have to a description of motivation is Plutarch saying he aimed to leave Italy by marching through the Alps.