Importunate has been part of the English language since the 16th century, and the synonymous importune arrived even earlier, in the 15th century. The seemingly superfluous inclusion of the suffix -ate in importunate is a bit mysterious; one theory is that English speakers modeled the adjective after words like obstinate. Importune and importunate come from Latin importunus. The prefix im- means "not," and importunus can be contrasted with Latin opportunus, which shares its meaning with and is the ancestor of our opportune, meaning "suitable or timely." The connection is obscure now, but opportunus itself harks back to the Latin phrase ob portum, meaning "[coming] to harbor." Importune, and later importunate, once meant "inopportune, untimely," but that sense is now obsolete.
Note: Compare Medieval Latin importūnātus and Middle French importuné in this sense, which are formally, though not semantically, past passive participles of the transitive verbs importūnāre and importuner.