Yes, as long as the image is CMYK it should still be CMYK in the output.
For example, here’s a ZIP with two JPEG images. The only difference is the color space; one is RGB (Sample.jpg) and one is CMYK (Sample_cmyk.jpg). I converted the RGB image to CMYK with a free online tool at https://www.rgb2cmyk.org/ but if you need to convert images you’re probably better off using Photoshop or Illustrator. As you can tell the colors are a bit off.
Sample.zip (2.2 MB)
To show that the CMYK image is output as CMYK, I placed both images in a blank template and created output. Adobe Acrobat shows that the image on the left has a DeviceRGB color space and the image on the right has a DeviceCMYK color space, as expected.