I suspect the white and black are both required elements of a QR code, otherwise it is out of spec.
K only colour image in psd
You have to make sure the background is transparent in Photoshop. So here's a QR code as a vector object on its own layer (there's no Background layer below) and you can see the transparency indicated by the check pattern.
Or if the background is white you can set the .psd to multiply over in ID, in the that case the black is going to overprint the color below. Here the top version is multiplied
Can you insert a screenshot?
(Also, is this related to InDesign? In InDesign you would not get to see a 'grey-white transparent area' – I presume you mean the Transparency checkerboard with that. Perhaps you accidentally asked here instead of in the Photoshop forum?)
in case there is no smart object involved, you could also do a mode change in PhotoShop:
A. To Grayscale (without transparency, only the background layer). Make sure, that the grayscale value is set to K: 100 for all drawing pixels.
B. To Bitmap
Save the image as TIFF and place that in InDesign.
When in InDesign you could recolor the TIFF image as you like.
Even set it to "overprint" or use the Effects Panel and set it to "multiply".
Example of two images placed in InDesign, a Bitmap and a Grayscale one.
On the right side after placing. Note: The grayscale TIFF image is already colored, but does not overprint!
On the left both images are colored with Cyan and both are set to "overprint".
Note: The image, not the container is selected.
Thank you for your answers. It seems that it is a sophisticated photoshop problem.
Looking at your captures in the Photoshop forum, it seems like you are trying to overprint the CMYK 0|0|0|100 black. To do that set the .psd to Multiply when you place it in ID. Here in InDesign you can see that when I hide the Black separation channel the black is overprinting the CMY underneath
It seems that the problem is solved. It took just a few seconds to change blending mode of an object in Indesign to multiply (and 3 minutes to find this option...). Thank you.