In German speaking countries, the term Wiener Klassik (lit. Viennese classical era/art) is used. That term is often more broadly applied to the Classical era in music as a whole, as a means to distinguish it from other periods that are colloquially referred to as classical, namely Baroque and Romantic music.
The term "Viennese School" was first used by Austrian musicologist Raphael Georg Kiesewetter in 1834, although he only counted Haydn and Mozart as members of the school. Other writers followed suit, and eventually Beethoven was added to the list. The designation "first" is added today to avoid confusion with the Second Viennese School.
Whilst, Schubert apart, these composers certainly knew each other (with Haydn and Mozart even being occasional chamber-music partners), there is no sense in which they were engaged in a collaborative effort in the sense that one would associate with 20th-century schools such as the Second Viennese School, or Les Six. Nor is there any significant sense in which one composer was "schooled" by another (in the way that Berg and Webern were taught by Schoenberg), though it is true that Beethoven for a time received lessons from Haydn.