The D810 requires at least Lightroom 5.6. The upgrade can be purchased from Adobe directly for $79. Here is the link:
You qualify for the upgrade because you have purchased Lightroom 4. After you install Lightroom 5 and start the program, it will see your Lightroom 4 catalog and will prompt you to allow it to create an upgraded version for Lightroom 5. This will create a new copy of the catalog, leaving Lightroom 4 completely intact and usable. You should be aware that Lightroom 5 is possibly nearing the end of its lifecycle. It is anticipated that Lightroom 6 will be made available sometime in the next few months, but that is only conjecture. There has been no announcement from Adobe. If you want Lightroom 6 when it becomes available it will be necessary for you to pay another upgrade fee. Depending on your requirements and plans, you may want to consider using the latest DNG converter to create DNG copies of your raw images for a month or two to see if anything develops from Adobe. On the other hand, if you think Lightroom 5 will be sufficient there is nothing wrong with purchasing it. It is a good program. You might also consider the creative cloud photography package ($9.99/month) which gives you the latest Lightroom and Photoshop. And when new versions of the program are released they will be made available to you automatically and at no additional cost other than your monthly subscription fee.