This is a fascinating question which taught me several new things about Roman culture!
The extent to which we can answer this question affirmatively depends heavily on how we define "sign language." I will divide my answer into three levels. I will exclude, as an obvious first level, the universal human ability to communicate via bodily attitudes, such as hand gestures.
Signs that indicate specific meaning (dactylonomy): Yes, the Romans had a well-known and attested numbering system involving the hands.
Signs that indicate letters (dactylology): Yes, certain classical authors seem to refer to this and Bede offers a comprehensive description of how to do this.
Signs that indicate words: I am not aware of a described system for this.
I will use many quotes from this article on antiquitatem.com for my answer. I recommend reading the whole article if you want to learn more.
Manual system for numbers
The Romans had a specific set of finger signs they used to count and calculate numbers. The system was much more refined than simply finger counting, which restricts one to 10): 1-9, 10-90 had specific representations on the left hand. 100-900, 1000-9000 had representations on the right. Combinations of these symbols (in succession, of course) could be used to represent numbers up to a million.
Juvenal uses this fact to explain how Nestor is more than a century years old:
Rex Pylius, magno si quicquam credis Homero, exemplum uitae fuit a cornice secundae. felix nimirum, qui tot per saecula mortem distulit atque suos iam dextra conputat annos, quique nouum totiens mustum bibit. (Juvenal, Sat X, 246 251)
Numerous other testimonies of this exist, from classical and Christian writers. The Venerable Bede (7th century) offers a comprehensive overview of this system in his De Computo vel Loquela Digitorum, which is really fascinating to read. As an example, here is his description of how to form "10" and "20" (on the left hand):
Quum dicis Decem, unguem indicis in medio figes artu pollicis. Quum dicis Viginti, summitatem pollicis inter medios indicis et impudici artus imittes.... (pg. 142)
Manual system for letters
This too is referred to (albeit less explicitly) in Roman times. Here is an example from Ovid:
Nihil opus est digitis, per quos arcana loquaris (Ovid, Ars amatoria, I, 137)
After describing numerical counting, Bede (op. cit.) describes how this same system can be used to spell words by individually indicating each letter's position in the alphabet:
Potest autem et de ipso quem praenotavi computo quaedam manualis loquela, tam ingenii exercendi, quam ludi agendi gratia figurari: qua literis quis singillatim expressis verba, quae iisdem literis contineantur, alteri qui hanc quoque noverit industriam...legenda atque intelligenda contradat.... (pg. 143)
It is perhaps significant to the spirit of this question that he sees this system as an "exercise of ingenuity" or a "game": this system is not well suited to communicating ideas efficiently.
He gives the following example:
Verbi gratia: Si amicum inter insidiatores positum, ut caute rem agat, admonere desideras: III, et I, et XX, et XIX, et V, et I, et VII, et V, digitis ostende. Hujus namque ordinis literae, Caute age, significant. (pg. 144)
He also indicates that this system is even easier with Greek letters, since by convention each Greek letter already has a numerical value.
As for a functional sign language, in the fullest sense, I cannot find any attestations.