If a monochromatic laser light of wavelength $\lambda$ is incident on a diffraction grating of period $A$, then does the second ($m$=2) diffraction orders have same or different $\lambda$? From the diffraction equation I see the wavelength, order and angle all variables so am not sure which will change.
The diffraction grating doesn't produce any output light with any wavelength that isn't present in the incoming beam.
However, you can analyze the different diffraction orders as resulting from the different spatial wavelengths present in the fourier decomposition of the transmission function of the grating.
For example, if you could make a grating whose transmission function was a perfect sinusoid as a function of position on its aperture plane, you would see only a single diffraction order. If you made a grating with a transmission function that's a perfectly periodic square function, you would get diffraction only into odd ordered beams (and the zero-order beam).