I have used the following imagemagick command for the image below:
convert img.png -define morphology:compose=darken -morphology Thinning Rectangle:17x1+0+0\< tmp.png
This removes ALL lines from the image, but I just want to remove the small horizontal and vertical lines on the right and bottom of the number in top left corner of each block. I want to preserve the main column and row lines. Can anyone tell me how to do it?

This is what I get (notice the long lines dividing the image content into columns and rows are also gone. I want those line to stay):
I notice the script is finer/thinner and less regular than the lines so it is more susceptible to eroding techniques. With that in mind we can ditch the text like this:
convert vcards.png -colorspace gray -threshold 50% -morphology erode disk:1.5 +repage z1.png

That's a good start, but if we use that as a mask, we will lose the long horizontal lines in your original image. So, we can find all those by projecting all the rows into a single-pixel wide tall column and thresholding all the rows that are more than 80% white. Then widen the image back out to its original width.
convert z1.png -colorspace gray -resize 1x\! +repage -threshold 80% -scale 810x1518\! +repage z2.png

Now combine the two masks so they only do the lower and right sides of your little title box things.
convert z1.png \( z2.png -negate \) -compose darken -composite z3.png

Finally, fatten that mask up a bit because it may have shifted around during previous processing, and apply it to your original image.
convert vcards.png \( z3.png -morphology dilate disk:2 -negate \) -compose darken -composite result.png

It could all be combined into a single command, but I won't do that, because some aspects may not work for all your images and while they are all individually implemented and documented, they are simpler to improve or correct individually.
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