You can export the work you have done on the laptop as a catalog, and include the images. You could export everything to an external hard drive. Then you could plug that external hard drive into the other computer and import from that catalog.
I am not sure I understand how to do this - do you mean as a collection?
On your laptop, you will need a way to choose which images you want to merge into your desktop catalog. A collection would be one way to do that. Folders named by date would be another option as would keywords or star ratings.
After you have selected your images to merge, from the File menu select "export as catalog" and choose the option to include the "negative files".
Copy that exported catalog and its files to your desktop using a network or external drive.
On your desktop, "import another catalog" to merge your images and edits into your main catalog.
Alternatively, you could put your images and catalog on an external drive, then move the drive from computer to computer as you need.
Thanks - I will give that a try.