Did you consider assigning values to the variables and plot x'(t); the intercepts on the x-axis will be your fixed points.

It should be easy to integrate x'(t) to get x(t). Then we need to assign some variable values.

variablesList = {G -> 1, Q -> 1, \[Alpha] -> 1, 
  x -> 1, \[Omega] -> 1, \[Gamma] -> 1}

Now plot the equation with the assigned values replacing the variables. I see at least one stable fixed point with the simple assignment.

Plot[
 (G/Q (2*BesselJ[1, v] + (\[Alpha] v - 3/4 \[Gamma] v^3)) - \[Omega]*
     x - v/Q) /. variablesList, {v, -4, 4}]

You should not separate the variable assignments with commas. Use semicolons or a list of rules. Hope this helps.

I am not familiar with the type of equation(s) above, but if you know v(t) and one initial condition, it is possible to find x(t) with DSolve.

DSolve[x'[t] == v[t] \[Omega] /. v[t] -> Sin[\[Omega] t], x[t], t]