I would go for no, not in the same way.

When dealing with "contemporary music" in normal situations, people tend to mix a lot of social issues with the actual music. Those issues are addressed separately when it comes to academic criticism (or at least an effort is made to try, in many cases).

Subcultures and generational gaps would not be comparable for the classical period. Did any people complain about their new music? Yes. Were parents or grandparents actively commenting on how they dislike this new music? I don't know, but there are instances of people disliking it, considering it incoherent, lacking proper structure, dissonant...

While not as far back historically as Mozart or Beethoven, you might find interest in the audience's reaction during the premiere performance of Stravinsky's "The Rite of Spring" in Paris (1913):

The music itself was angular, dissonant and totally unpredictable. In the introduction, Stravinksy called for a bassoon to play higher in its range than anyone else had ever done. In fact, the instrument was virtually unrecognizable as a bassoon. When the curtain rose and the dancing began, there appeared a musical theme without a melody, only a loud, pulsating, dissonant chord with jarring, irregular accents. The audience responded to the ballet with such a din of hisses and catcalls that the performers could barely hear each other.

Backstage at the premiere, Nijinsky shouted at the dancers while Diaghilev tried to suppress a possible riot by flashing the house lights. Stravinsky himself fumed at the audience's response to his music. Source

The scope, intensity, and even the cause of this "possible riot" has been debated, and we may never know all of the exact details. To whatever extreme it may have escalated, it leaves us with the impression that it was possible for the "modern music" of the day to stir up some pretty negative reactions from people.

Skepticism of new music has certainly been a common occurrence throughout music history and was not reserved for older generations. It is important to remember that attending the symphony and especially the opera during the 18th and 19th centuries was a main form of entertainment for many classes of people (not just the elite). As in any aspect of social culture, change was not always accepted right at first. The example of "The Rite of Spring" premiere is a good example of this. In terms of Mozart for example, his music did not instantly surpass those of his contemporaries such as Antonio Salieri, despite popular conception based on the movie "Amadeus." Salieri's music was familiar to the music-going public and he remained just as, if not more popular than Mozart during their lives. Of course Mozart proved to survive in the western art music canon better than Salieri, but opinions of him were not exactly the same as they are today when he was alive.

Beethoven similarly was not always immediately accepted, largely based on his innovations in form. For example, Beethoven was the first to replace the traditional third movement "minuet" in symphonic form with a "scherzo." While later composers regarded Beethoven as a sort of superhuman musician who set the highest standards in composition, his audience certainly had to get used to his changes in music.

This is more of a one-off story than a real trend, but it might interest you. Joseph Haydn became employed by the Esterhazy family as Classical music was rising to prominence over the older Baroque style. The man Haydn would eventually replace as Kapellmeister, Gregor Werner, was an older, firmly Baroque composer, and cared for neither Haydn nor the simpler, lighter style of composition that he represented.

This is not an answer, but an attempt to clarify, since we have opposite indicating responses right now. Vague memories from Music History classes say that Bach was criticized for using tempered harmonics, or maybe it was minor keys. In any case, it was called 'left-hand' or devil's music. This sounds just like the kind of accusation of music we hear today, so it seems valid. But I can't recall more details than this and I'm hoping that it might prompt someone more knowledgeable... Anyone? Was Bach accused of using satanic music?