We present AdaptDQC, an adaptive compiler framework for optimizing distributed quantum computing (DQC) under diverse performance metrics and inter-chip communication (ICC) architectures. AdaptDQC leverages a novel spatial-temporal graph model to describe quantum circuits, model ICC architectures, and quantify critical performance metrics in DQC systems, yielding a systematic and adaptive approach to constructing circuit-partitioning and chip-mapping strategies that admit hybrid ICC architectures and are optimized against various objectives. Experimental results on a collection of benchmarks show that AdaptDQC outperforms state-of-the-art compiler frameworks: It reduces, on average, the communication cost by up to 40.2% and the latency by up to 38.4%.