There is an arrival for James Lindsay June 30, 1903 on the Sicilian sailing from Glasgow to Montreal. He is 23, from Lanark, Scot, occupation - traveller, Canadian destination Calgary NWT - no indication as to marital status. A possibility but perhaps not a strong one.
Other sailings for this family include
James Lindsay travelled back from Greenock to Quebec arriving 10 Sep 1927 on the Montrose
James and Annnie travelled back from Greenock to Montreal arriving 23 Oct 1931 on the duchess of Atholl
Annie (50) and daughter Maud (22) b. Sask. travelled back from Greenock to Quebec arriving 21 Oct 1932 on the duchess of Bedford. Annie indicated she had been in Canada from March 1905 to July 1932.
Annie G.S. and daughter Mary (11) b. Limerick, Sask. travelled from Glasgow to Montreal arriving 3 Sept 1926 on the Metagama. Sister, Miss Roberts 7 Glenbank Terrace Lenzie Scottish contact. Again Annie indicated her stay in Canada started in 1905.
The family, James, Annie, Charles Stewart, James, Mary, Maud and David Wyle, travelled back from a family vacation on the Tunisian arriving Sept 1921. Indication that the family arrived in 1914 into the port of Boston.
There is a record for Mrs. Constance (James) Lindsay with David Wylie and Maud arriving 1 April 1910 from Glasgow to Boston
David Wylie Lindsay, travelled on the Duchess of Atholl from Greenock to Montreal arriving 19 July 1929.
UK inboard records include
Annie G. Lindsay, b. 28 Nov 1881, Montreal to Greenock on Empress of Britain arriving 9 June 1958 with contact Miss Steward 73 Glenbank Rd Lenzie Scotland
Annie C. and Maud (stenographer) travelled from Montreal to Glasgow arriving 6 Aug 1932 on the Montcalm
Mary and Annie C. travelled from Montreal to Glasgow on the Megagama arriving 30 May 1926
James and Annie travelled from Montreal to Greenock arriving 15 Aug 1932 on the Montclare
Annie G. Lindsay, b. 28 Nov 1881, travelled on the Empress of Scotland from Montreal to Greenock arriving 30 Apr 1957.
David Wylie 22, student travelled from Montreal to Liverpool on the Duchess of Atholl arriving 12 Oct 1929
David Lindsay, 16, student, travelled on the Metagama from St. John NB to Glasgow arriving 10 Feb 1924
The family travelled a lot. Still didn't find the answer to your original question of their arrival.
Debbie