In the absence of film footage and accurate timing, it is impossible to say definitively, but based on numerous newspaper reports these are Scotland's fastest:
Archie Robertson v Austria, 19 May 1955. According to several reports including the Evening Express, he scored after just 15 seconds; the Evening Times said 20 seconds. Scotland went on to win 4-1 in Vienna on their end-of-season tour.
Andy Black v Czechoslovakia, 8 December 1937. Most papers, including the Evening Express, Daily News and Evening Despatch all agree 20 seconds. And what is more, Black was making his international debut! This was the opening goal of a 5-0 win at Ibrox.
The record appears to belong to Robertson, but Black makes a strong claim.
Stevie Chalmers v Brazil, 25 June 1966. In a match best remembered for Pele's only appearance at Hampden, Chalmers fired home the opener in the first minute. Timings vary, with the Sunday Sun saying 30 seconds, the People judging it 45 seconds, and one fan who was trying out a new stopwatch recorded it at precisely 38 seconds. The game ended 1-1.
RS McColl v England, 7 April 1900. Wearing Rosebery strips, the opening salvo of a first half hat-trick by 'Toffee Bob' came after 'less than 40 seconds' according to the Evening Standard, 'in forty seconds' thought the Belfast News-Letter, while the Lancashire Evening Post said 'at the end of 45 seconds'.
John White v West Germany, 6 May 1959. This goal in a 3-2 victory was reported in most British papers as 'the very first minute' but I found a more precise time of 50 seconds in a German paper, Fussball Sport.
England captain Bob Crompton managed to score an early own goal for Scotland at Newcastle on 6 April 1907 which was timed variously as 'less than a minute' (Empire News), 'scarcely a minute' (Sports Argus), 'in the first minute' (Morning Post) and 'exactly sixty seconds' (Scottish Referee).
Willie Lambie v Ireland, 30 March 1895. His opening goal at Celtic Park was described as 'less than a minute' (Dundee Advertiser and Belfast News-Letter) and 'in the first minute' (Morning Post), while others thought 'in a minute', 'only a minute', 'about a minute' and 'hardly a minute'.
The quickest in the current century appears to be Billy Dodds v Belgium on 24 March 2001 which was timed at one minute and ten seconds.
Needless to say, Scotland has also conceded some quick goals with two prime candidates for fastest:
Edgar Chadwick (England) on 2 April 1892 was timed at 30 seconds by the Lancashire Evening Post and 'less than a minute' in the Evening Standard.
Dave Walsh (Northern Ireland) on 17 November 1948 was also clocked at 30 seconds in the Daily Herald, while other papers thought it was 40 seconds.
More recently, Robbie Earnshaw's first of a treble for Wales on 18 February 2004 was timed at 42 seconds on Sky Sports and 43 seconds on the BBC.
Denis Wilshaw (England), 2 April 1955, scored in 45 seconds according to the Evening Times and Belfast Telegraph.
And finally, Georges Aeby of Switzerland opened the scoring at Hampden on 15 May 1946 'in the first minute' according to most papers, but nobody gave a more accurate time. Similarly William Kenyon-Slaney scored the first ever international goal for England on 8 March 1873 in one or two minutes, but none of the reports was concerned with precise timings.
That appears to be all the goals scored in the opening minute of Scotland matches, although it is possible that evidence will be found to exclude these candidates, or even to add another to the list.
NB I have only covered the Scotland men's team above, as few detailed records exist for the women's equivalent. However, it should be mentioned that Julie Fleeting twice scored in the opening minute, v Estonia on 3 September 1997 and v Portugal on 17 February 2001. So did Hayley Lauder v Israel on 16 June 2012, after 58 seconds. However, neither of them can beat Kathryn Morgan of Wales who scored against Scotland in just 13 seconds on 2 June 1996.