J[[Omega]_] =  1/Sqrt[2 [Pi]] ([Lambda]^2 [Gamma])/(([Omega] - [Epsilon])^2 + \[Lambda]^2);

f[t_] = Assuming[{[Lambda] > 0,t > 0, [Epsilon] [Element] Reals, [Omega] [Element] Reals, [Epsilon] > [Omega]},InverseFourierTransform[J[[Omega]], [Omega], t]] // FullSimplify

FourierTransform[f[Abs[t]], t, [Omega]]

([Gamma] [Lambda] (I [Epsilon] + [Lambda]))/(Sqrt[2 [Pi]] (-[Epsilon]^2 + 2 I [Epsilon] [Lambda] + [Lambda]^2 + [Omega]^2))