Hey, I'm using an HP Z400 right now. To your question, no this should not lead to problems. Unless the GTX GPU has a higher recommended wattage than your PSU can output. What GTX 1060 are you planning on getting? I'm working on upgrading my PSU in my Z400 right now. Wich can be found at the link below.
http://www.tomshardware.com/answers/id-3865109/finding-psu-power-supply-unit-z400.html