Well, if you compare two scenarios:
a) using a 1.3x longer lens with no need to crop and hence using the whole sensor, versus
b) using a shorter lens and using 1.3x crop mode, then
the cropped image needs to be enlarged more and that brings out the noise. Also the cropped sensor received less light, and the amount of light is what determines noise.
In the end you'll end up with about 3/4 stop more noise with the crop.
The math is: log(1.3 x 1.3) / log(2) = 0.757 stops, or about 3/4 stops more noise.
And you have the camera, hence no need to rely on armchair ponderings. Go out with a zoom lens and see for yourself.