By default, no, however, there is a feat from the Weapon Master's Handbook, Slipslinger Style that allows the Warslinger racial trait to work with the Halfling Sling Staff:
Slipslinger Style (Combat, Style)
Prerequisite(s): Weapon Focus (sling) or weapon training (thrown) class feature, warslinger racial trait
You gain a +1 bonus on damage rolls made with all types of slings, and don’t provoke attacks of opportunity when reloading a sling. In addition, you treat all ranged and thrown weapons that have “sling” in their name as if they were slings for all feats and class abilities that require such weapons, as well as the warslinger racial trait
(emphasis mine) The Halfling Sling Staff is an exotic ranged weapon with the word sling in it, so the feat applies.