I want to know whether Compton effect depends on the frame of reference or not. The wavelength change is a thing, but does not depend on frame of reference since it depends only on $\theta$. But if I travel at a speed comparable to velocity of light, will there be any changes?
Yes, the details of Compton scattering are frame dependent.
The usual Compton scattering formula assumes a target electron at rest in the frame of the observer.
If the observer and electron have significant relative motion, then a couple of Lorentz transforms are required to put the incoming photon in the rest frame of the electron, then to put the scattered photon back into the frame of the observer. This changes the calculated photon energies and scattering angles, and can result in the scattered photons gaining energy from the electrons (a.k.a. inverse Compton scattering).