The plot *is* showing this behavior. Change the PlotRange to see it more clearly.

    Plot[Q[\[Omega]], {\[Omega], 0, 3}, 
     PlotRange -> {{0, 0.05}, Automatic}, PlotStyle -> {Black}]

Better still is to check the limit of the function as \[Omega] -> 0

    In[6]:= Limit[Q[\[Omega]], \[Omega] -> 0]
    
    Out[6]= 0