No Klimt represented the "Secession" movement which at that time was in opposition to the establishment and the academy. According to wikipedia: "in particular, opposing the domination of the official Vienna Academy of the Arts, the Vienna Künstlerhaus, and official art salons, with its traditional orientation toward Historicism. " https://en.wikipedia.org/wiki/Vienna_Secession

I think Nazims famous opposition to modern art have been conflated with Hitlers own rejection from the academy. He wasn't rejected due due different genre preference, but because he wasn't good enough in drawing anything beside architecture.