The Gunners have been unable to hold aloft any major trophy since the 2005 FA Cup and Adams, who won four league titles, three FA Cups, two League Cups and a UEFA Cup Winners' Cup during his playing days, is hoping to see that change.
"The fans are desperate for a trophy of some sort," he told Sky Sports.
"I would like the players to get used to winning; if you have never had that experience of crossing the line as a winner you don't know what it is like to be a winner."
And to so, Adams has targeted some areas that manager Wenger needs to improve.
"Lately Arsenal have become very solid and (Laurent) Koscielny and (Per) Mertesacker have gained a confidence together, but defensively I think they are still short - I'd like to see them sign another centre-back, a couple of full-backs and possibly a keeper," he added.