In general, yes. The Romans had contact with the Greek cities of Magna Graecia from a rather early date, and they had extremely close contact with the Etruscans even earlier. The Trojan Cycle appears in Etruscan art from a very early date, and the Etruscans traded extensively in Attic pottery, so mythological motifs are quite common even when they were not necessarily prominent in Etruscan or Western Greek cults. Mind you, the early Romans would've been exposed largely to the Etruscan and West Greek cults, which are not necessarily the ones moat familiar to us--for example, the cult of Aeneas was apparently quite prominent in Etruscan tradition but is rather absent from most Greek traditions. The Homeric Poems were almost certainly known from an early date, but the earliest we can be positive that the poems themselves were known (rather than the stories in them or different sets of poems in the Epic Cycle entirely) is Livius Andronicus' translation of the Odyssey, but surely the existence of the poems was well-known before then, given its ubiquitous prominence in Greek tradition, even if it could not be read in Latin (and knowledge of Greek was not particularly widespread among Romans before Scipio popularized Hellenism). But elements of Greek cultic traditions, either through the Etruscans or directly through the West Greeks, appears from pretty early on