To a certain extent, yes. For music, at least, there were forms of awards, but they would usually be something like court appointments - being the official composer, musician, etc. to the King, Prince, Duke, whoever. (I'm sticking to music for this answer, because that's obviously my thing) These were highly sought-after because it meant a steady income from a very wealthy patron, and not having to constantly find commissions or take students in order to make ends meet, so having a lucrative court appointment was both prestigious and secure. Joseph Haydn was Kappellmeister to the wealthy Esterhazy family, Bach was Cantor of St Thomas Church in Leipzig (among others), and Henry Purcell was Organist for Westminster Cathedral and the Chapel Royal at the same time. While these positions sound like they were employed to play an instrument or teach, a great deal of composition was part of the positions as well, and many of their greatest and most prolific works were composed during these appointments.

You'd also have composers get knighted, Mozart for example was knighted by the Pope, Arthur Sullivan (of Gilbert and Sullivan fame) was knighted by Queen Victoria, and Farinelli, one of the most famous castrato singers was knighted, among many others.

So the answer is kind of? There were awards, but they weren't the same as things like the Academy Awards of today.

I am not at all an expert on this subject, but it's been four days now, so since probably no one else will, I might as well bring up the Dionysia. A yearly contest was held in classical Athens as part of the festival honoring Dionysus, in which new plays--at first only tragedies, but later comedies as well--were performed, and a single playwright was crowned the victor for their dramatic talents.

Plato's Symposium also briefly mentions another such contest, the Lenaia, which one of the dialogue's main figures has just won.