It's simple. Lena is 512 x 512 pixels with a bit depth of 8. So you simply export the raw image to a file that will be exactly 262,144 bytes long. It's critical that you export or save the image as a raw binary file, not one of those JPEGey things. That won't work.
To determine the two chi values, you then consider the file as a series of consecutive individual bits, or as pairs of bits. This will entail a load of bit shifting /masking operations in your favourite programming language.
A tip. Install ent, and just run it twice against the encrypted Lena file. Run one, use the -b option which will treat the file as a series of bits. The output will produce a chi and p value. Run again without the b option. This will process the file as a series of 8 bit values. If the encryption produces independent (non correlated) bits, they can be lumped into four pair groups and treated as octets. Assume the same 5% confidence value, and look at the returned chi value to see if it's acceptable. You'd expect $219 \leq \chi^2 \leq 293$ for randomly distributed octets.
ent will also give you the Shannon entropy of the images for comparison with those in the paper.
Another tip. Do not assume that a chi failure means the encryption fails. Randomness is pesky and you have to try the test several times on repetitive images or keys.