The plot is showing this behavior. Change the PlotRange to see it more clearly.
Plot[Q[ω], {ω, 0, 3},
PlotRange -> {{0, 0.05}, Automatic}, PlotStyle -> {Black}]
Better still is to check the limit of the function as ω -> 0
In[6]:= Limit[Q[ω], ω -> 0]
Out[6]= 0
