Since x and y are Real, ComplexExpand may be able to help
a=3;b=0.0137;c=0.0023;Q=6;
NIntegrate[ComplexExpand[Im[Exp[-I x c+b (I x y/(y^a-I x))]]/x], {x,0,Infinity},{y,0,Q},AccuracyGoal->10]
returns
-8.79547
but warns it was not able to obtain 10 digits of accuracy near x==0.
Dividing the integral into two parts
a=3;b=0.0137;c=0.0023;Q=6;
NIntegrate[ComplexExpand[Im[Exp[-I x c+b(I x y/(y^a-I x))]]/x],{x,1/10,Infinity},{y,0,Q},AccuracyGoal->10]+
NIntegrate[Im[Exp[-I x c+b(I x y/(y^a-I x))]]/x,{x,0,1/10},{y,0,Q},AccuracyGoal->10]
returns
-8.79548+0. I
with no warning, but I would study the accuracy of that carefully