Most likely, yes, they did. Here is an old post of mine which answers your question as well as gives a bit of more background on how the best-known Roman origin story developed.

I'll copy and paste it here for convenience:

Are you asking about the legendary origins? In this case, the Romans were descended from both the Latins and the Trojans. According to this popular variant of the foundation myth, Aeneas the Trojan flees to central Italy. There he marries the daughter of local king Latinus. Other Trojans also marry the locals, and their progeny are called the Latins. Romulus and Remus are direct descendants and found the city of Rome. Therefore, the Romans were descendants of these Latins, who were themselves descended from Trojans.

That is the simple, established version. But we are probably dealing with two separate mythical traditions here which were joined together at a later time. One involves Aeneas the Trojan, and one involves Romulus the Latin. The popular version I gave above is the final result of a long process.

The idea that the Romans were descended from the Trojans was very old and originated with the Greeks. The earliest references we know of are from the end of the fifth century, when both Hellanicus of Lesbos and Damastes of Sigeum claim Rome was founded by Aeneas of Troy. However, it is possible the legend was already established in central Italy a century prior to this.

Another, separate myth had Romulus as the founder of Rome. This is the popular Romulus and Remus and the she-wolf story everyone knows. This was probably an indigenous legend and did not come from the Greeks or Etruscans or anyone else. It was a homegrown foundation story. This was at least as old as the Aeneas story, because the famous she-wolf bronze statue has been dated to the mid-sixth century.

Over time the Aeneas version of the myth became more and more popular around the Mediterranean, particularly when Rome grew in size and influence and had more dealings with the Greek world. One idea is that Aeneas really exploded in popularity in Rome by the time of the war with Pyrrhus around 280 BC. Pyrrhus claimed descent from Achilles and portrayed his attack on Italy as a new Trojan War. This meant the Romans were the Trojans, which the Romans happily accepted.

But there was a problem. If the Romans wanted to believe they were Trojans, what were they to do with their local myth about Romulus? Simple. Romulus became Aeneas’ son or grandson.

But there was another problem. According to calculations from Greek scholars, the Trojan War probably took place around 1200 BCE in our reckoning. But the Romans already had established a timeline with Romulus being alive in the eighth century. How to fix this? Simple. Insert a long line of Kings of Alba Longa between Aeneas and Romulus.

And with these little tweaks, the Romans could be descended from both Aeneas and Romulus. So in the end, the Romans counted both the Trojans and the ancient Latins as their ancestors.

My favourite source for this is T.J. Cornell’s The Beginnings of Rome.

Also, note this part of the old thread which I linked to:

In Rome’s case it is notable that by accepting the Aeneas myth they were calling themselves Trojans, not Greeks, and so put themselves into opposition to the Greek world. This was probably useful for the contemporary situation as they expanded. They were able to put the current wars against Greek people into a mythological-historical context.

This upset some people such as Dionysius of Halicarnassus. He was a Greek writing for a Greek audience and tried to claim that the Romans were not Trojans but were instead Greeks. At the time of his writing Rome was rapidly expanding and had established hegemony over much of the Greek world. Dionysius was trying to reassure the Greeks that they had not been beaten by their old enemies; instead the Romans were Greek just like them.