Do you have an option to export to XMP? This will write XMP sidecar files in the originals' folders. Move them to the other computer with your originals, then import the originals into LR.
Alternatively, ensure use the Export / Import Photo Binders feature to move your pictures.
Thirdly, if your new computer has the pics in the same folder names and structure as the other computer, then copy the LR library folder to the new computer.