To a small but real degree, yes. Higher-quality wire uses high-purity copper with lower conductor resistance, so less energy is lost as heat as current flows. Over years of use across a whole home, that reduced loss can translate into modest energy savings — and cooler-running wire also lasts longer and is safer.
The effect is greatest on circuits that carry heavy loads for long periods, such as air-conditioners and geysers, and where wires are correctly sized rather than undersized. It is not a dramatic saving, but it is a genuine bonus on top of the main benefits of safety and durability.
Look for pure electrolytic copper and energy-saving claims backed by real certifications. APAR markets low-conductor-resistance, energy-saving house wires for exactly this reason.