Hello
You made a syntax mistake.
Sin[\[Omega] t] == q''[t]/Subscript[dw, 0]^2 + (2 \[Xi]q'[t])/Subscript[dw, 0] + q[t]/d
It should be:
Sin[\[Omega] t] == q''[t]/Subscript[dw, 0]^2 + (2 \[Xi] q'[t])/Subscript[dw, 0] + q[t]/d
You forgot a backspace to add between
$\xi$ and q'(t)
NSolve is a numerical solver, can't solve a symbolically.
Clear["Global`*"]
eq = Sin[\[Omega] t] == q''[t]/Subscript[dw, 0]^2 + (2*\[Xi] *q'[t])/Subscript[dw, 0] + q[t]/d
eq1 = LaplaceTransform[eq, t, s] /. q[0] -> 0 /. q'[0] -> 0 (* Initial conditions,you may delete it or change to: eq1 = LaplaceTransform[eq, t, s] *)
sol = First@Solve[eq1, LaplaceTransform[q[t], t, s]] // Simplify
q[t] = InverseLaplaceTransform[LaplaceTransform[q[t], t, s] /. sol, s, t]