I bought LR4 then upgraded to 6 in June. But yesterday I finally upgraded on the desktop. The assigned serial # will not work on the laptop!!!
Let us know the exact error message you are getting while doing the activation on your laptop.
You can install one license on two system at time, but one of them at a time.
Refer the below link and download Lightroom 6 on the laptop and enter the upgrade Serial #. Then it will ask you to select the base product, select Lightroom 4 and enter the serial #.
Let us know if that helps.