As not unusual with models from this period, opinions differ ...
Of the twelve published sources I've looked at:
6 say 1964
5 say 1965
1 says 1966.
Hmmm ...
Dealing with the 1966 date first. This comes from
Motociclismo and is simply explained by the fact - and everyone agrees on
this point, at least

- that the model was released late in Italy. So we can discount this date.
So, 1964 or 1965? Well:
Two of the leading Ducati authors, Mick Walker and Ian Falloon, are themselves inconsistent in what they say, Ian Falloon even going so far as to claim both years on the same page in
Ducati Overhead Camshaft Singles 
. Of the others, Tom Bailey says 1965, Alan Cathcart says 1964. And then there's the danger that lesser authors just repeat what they read, of course ...
So back to Ducati literature:
There is an owners handbook dated July 1965. This
claims to be the first edition. BUT the model illustrated is the
second version of the Monza Junior (square tank, long side panels). So not convincing.
Ah ha! There was an earlier "
Supplement to Instruction manual, Over Head Camshaft 160 MONZA Jr" (sic). And it's date?
December 1964.
So that's what I would go for! But I may be wrong, of course ...

One way to check would be to go thru some bike magazines of the period and see what what being advertised for sale at the time.