The concept of a 'golden age' is fairly new and usually describes a period long after the fact. Usually a period with a large impact on the whatever it's the golden age of.

There are a few points in history one could point to as golden ages in Germany. Defining Germany however, can be quite difficult as it's only existed as its own state under that name since 1871. Relics of early Germanic/Frankish rule have been used in ceremony and as national symbols in Germany, Austria and France (referring specifically to the personal effects of Charlemagne). The national-identity we think of as German didn't exist until much later in history. That said what is now Germany, and people who have spoken German have had their time in the limelight. The histories of great kingdoms, such as the Holy Roman Empire of the German Nation, the Kingdom of Prussia, and the houses Hohenzollern, Hohenstaufen, Habsburg and of Hanover (which included 7 monarchs of the Great Britain and the United Kingdom one of which being Queen Victoria) are so vast I couldn't possibly do them any justice here (the Holy Roman Empire lasted 1006 years from 800 to 1806). At various points in history German monarchs controlled land from what is now Sicily, France, Poland, the Czech Republic, Denmark, the Netherlands, Belgium, Switzerland, Austria, Hungary and so on.

The 18th and 19th centuries, stretching into the 20th as well, are a good example of what one could call a golden age. In this period Germany (again, being loose with the word Germany) became an international power, with Frederick II of Prussia and Friedrich von Steuben influencing the American war of independence and further Prussians liberating much of Western Europe from Napoleon. Germany also experienced a cultural renaissance, with German romanticism, Sturm und Drang and Weimar classicism bringing forth many of Germany's most famous artistic and literary characters such as Johann Wolfgang von Goethe, Friedrich Schiller, Friedrich Nietzsche and Karl Friedrich Schinkel. The aforementioned city of Weimar was the center of much of this, along with Vienna, Berlin and a few others. Germany was also unified during this period by Otto von Bismarck.

Germany was quite dominant during the classical period of musical history, producing Mozart, Beethoven, Bach, Schubert, Haydn, etc., a legacy that would carry on into the history of Germany the nation with composers such as Richard Wagner and Johannes Brahms. You may or may not know the names or works of some of these men but if you were to listen to a playlist of their songs you would recognize a good many. German culture would also seep into the famous works of other nations such as the prolific work of Tchaikovsky, who was often criticized for sounding 'too German' (or at the very least non-Russian).

An exhaustive list of great German scientists would require a book unto itself. A few that you may know: Karl Gauss, Robert Bunsen, Emil Erlenmeyer, Albert Einstein, Wernher von Braun, Werner Heisenberg, Karl Bosch, Max Planck, Hermann von Helmholz, Heinrich Hertz and the list goes on.

Germany has been called 'das Land der Dichter und Denker', the 'land of poets and thinkers'. German ideas such as those of Leopold von Ranke, who set the standards of modern historical analysis and study, are core to this very subreddit. The Prussian virtues are still cited to this day from Germany to Chile (if you've never read them they're pretty much a list of every stereotype about German culture). German fairy-tales such as Rapunzel and Rumpelstiltskin are known the world over and like much of the most famous German music, come from the early 19th century and were originally brought together by the Brothers Grimm.

Interestingly enough Germany's most famous castle, Neuschwanstein, which the Disney castle was made to look like, was also built in this period, albeit in an older style and as a part of the romanticist movement.

There's just so much that happened in this period I've probably forgotten something, but that just goes to show my point.

EDIT: I slept on it and yes, I forgot a good many import things.

Most important of which being the economy of Germany in this period. The 1800s saw the German economy evolve from a primarily agrarian one to the industrial powerhouse we know it as today. By the turn of the 20th century Germany's economy has eclipsed that of the British Empire, which was at its peak. This was in part due to external forces (read: Napoleon axing many restrictive economic policies during his occupation) but also included internal action to cut the red-tape, taxes and tariffs that held the economy back, cash injections to promote growth in the steel and coal industries, and people like Helmuth von Moltke promoting and developing Germany's infrastructure, particularly railways. This revolution led to areas like the Rhein-Ruhr Gebiet becoming industrial powerhouses in Europe and became a great source of national pride. I live around there and to this day people speak a dialect of German peppered with words from the heyday of the coal and steel industry (for example, no matter where they work they say they're 'going up the mountain' instead of 'going to work') called Ruhrdeutsch.

And how could I have possibly forgotten Karl Marx and Friedrich Engels. Marx wrote so many books I laugh every time I hear a high-schooler or college-freshman say they've read enough to say they know what he was talking about and is one of the most important modern political figures, period, having penned the Communist Manifesto with Engels in 1848.