but does the hardware affect the output quality/size if the settings are identical?

Yes, your laptop has a much newer quicksync block with proper HEVC hardware-

NAS is Gemini Lake, which uses QS from Broxton/Apollo Lake/Gemini Lake

Laptop is Tiger Lake, which uses about 3 versions newer QS, for Tiger Lake, Alder Lake and Raptor Lake.

Broxton QuickSync doesn’t actually have an asic encoder path for HEVC, it’s all via shaders.