Subject to the huge semantic gap between natural and formal languages, neural semantic parsing is typically bottlenecked by its complexity of dealing with both input semantics and output syntax. Recent works have proposed several forms of supplementary supervision but none is generalized across multiple formal languages. This paper proposes a unified intermediate representation (IR) for graph query languages, named GraphQ IR. It has a natural-language-like expression that bridges the semantic gap and formally defined syntax that maintains the graph structure. Therefore, a neural semantic parser can more precisely convert user queries into GraphQ IR, which can be later losslessly compiled into various downstream graph query languages. Extensive experiments on several benchmarks including KQA Pro, Overnight, GrailQA, and MetaQA-Cypher under standard i.i.d., out-of-distribution, and low-resource settings validate GraphQ IR's superiority over the previous state-of-the-arts with a maximum 11% accuracy improvement.