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

When you simultaneously cross-post the same question (https://mathematica.stackexchange.com/questions/229514/numerical-integration-of-double-integral-with-two-variables), you should mention that in both posts.

With the code change above, I do not get the non-numerical error. However, the NIntegrate reports the integral as not converging. Dealing with convergence of multidimensional integrals is tricky. It could be that the integrand is oscillatory or contains singularities, or that it in fact does not converge.