I would start by checking whether the licence is linked to a Microsoft account, because that decides how I move it. A one-time purchase of Office Home can be used on one PC at a time, so I would uninstall Office from the old PC before I activate it on the new laptop.
On the new laptop, I would sign in to my Microsoft account and open the Services and subscriptions page. If Office Home is listed there, I would install it from that page and sign in with the same account when an Office app asks me to activate it.
If it is not listed under that account, I would redeem the product key on the Office setup page while signed in with the account I want to keep it under, then install from there. I would check the account first, because a key redeemed under a different account will not show up under this one.
If activation still fails, I would note the exact error message. I would also check whether the licence came preinstalled with the old PC, because an OEM licence is tied to that PC and cannot be moved. In that case I would need a new licence for the new laptop.