The earliest use of Golden Age with reference to musical theatre in general which I've been able to find is in this May 3rd 1977 New York Times review by Clive Barnes of a revival of The King and I with Yul Brynner. The title is of the review is:
'King and I,’ Reminder of Golden Age
The review goes on to call The King and I
a reminder of what the Broadway musical was like in its Golden Age of
This New York Times review is mentioned in Gerald Boardman's 900+ page tome American Musical Theatre: A Chronicle, first published in 1978. He makes one other passing reference to Golden Age in relation to musical theatre, but otherwise reserves the term for The Golden Age of the American Musical, 1924-1937 as one of his chapter (or 'Act') headings.
William E. Everett's The Musical: A Research and Information Guide cites hundreds of sources, including many which use Golden Age in the title, but none of these pre-date 1977.
There is an earlier use of Golden Age with a specific reference to Rogers and Hammerstein, the team generally considered to have initiated the Golden Age, starting with the 1943 Broadway production of Oklahoma!. David Ewan, in New Complete Book of the American Musical Theater (1970), writes:
The golden age of Rogers and Hammerstein ended with the Sound of
Ewan also uses golden age with reference to American comic operas and operettas.
As noted by the OP, there are different dates attributed to the Golden Age of Musical Theatre. Almost all agree on 1943 as the starting point, and most give the late 1960s (especially 1968, example here) as the end point. Some, though, give other dates (e.g. 1959, see here).
(all emphasis is mine)