Program obfuscation aims to hide the inner workings of a program while preserving its functionality. In the quantum setting, recent works have obtained obfuscation schemes for specialized classes of quantum circuits. For instance, Bartusek, Brakerski, and Vaikuntanathan (STOC 2024) constructed a quantum state obfuscation scheme, which supports the obfuscation of quantum programs represented as quantum states for pseudo-deterministic quantum programs with classical inputs and outputs in the classical oracle model. In this work, we improve upon existing results by constructing the first quantum state obfuscation scheme for unitary (or approximately unitary) quantum programs supporting quantum inputs and outputs in the classical oracle model. At the core of our obfuscation scheme are two novel ingredients: a functional quantum authentication scheme that allows key holders to learn specific functions of the authenticated quantum state with simulationbased security, and a compiler that represents an arbitrary quantum circuit as a projective linear-plus-measurement quantum program described by a sequence of non-adaptive Clifford gates interleaved with adaptive and compatible measurements.