The Scots Peerage, vol. 8, p. 168 has William's wife as Elizabeth daughter of Alan Coutts, Cutes or Couttie, based on the will of Alexander Alexander, William's eldest son, who names his guidschir (translated as grandfather) as Alane Cutes or Cutis.
The Memorials of the Earl of Stirling does have the will in full, and you can see the reference to Alane Cutis five lines from the bottom on page 27 but as stated in The Scots Peerage, the author of the Memorials interprets this as father-in-law, and has Alexander's wife as Marion daughter of Allan Couttie, even though the will back at the top of page 27 mentions Marion Graham as his spouse.
(The Memorials book actually has quite a few differences with The Scots Peerage, so it may not be very reliable.)
It is thought that Elizabeth is the daughter of Alan Coutts and his wife Marjorie Walwod, and the document that confirms Marjorie is Alan's wife, dated to 19 Oct 1555 also mentions William Alexander, but my latin isn't very good, so I'm not sure in what capacity. See here document no. 1005.