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.
