Smoker box for gas grills is what you seek..
like this.
HERENothing more than a small box which holds wood chips, has a lid and/or sides with small perforated holes to limit the oxygen to prevent flames but allow the wood to smolder and smoke..
I have done something similar to this with a simple homemade metal box from scrap leftover sheetmetal (not galvanized please), left it open top. Took some small tree twigs and a few small splits of Maple and start it smoking with a torch..
Green wood however typically is difficult to light and the smoke will be much stronger to the point of being harsh "licking a used stove pipe" creosote taste if you are not careful. I also prefer less bark as it can cause some off flavors..
There are a lot of different ways to generate a good tasting smoke on a gas grill, Ama-zen smoking tubes are a good choice for smoking pellets (looks simular to pellets for pellet stoves but does not have artificial binders).
Smoking boxes are designed for wood chips but I suspect if you have small enough chunks/chips/kindling/splits of Mesquite it might work.
