The First Passage Time (FPT) is the time taken for a stochastic process to
reach a desired threshold. It finds wide application in various fields and has
recently become particularly important in stochastic thermodynamics, due to its
relation to kinetic uncertainty relations (KURs). In this letter we address the
FPT of the stochastic measurement current in the case of continuously measured
quantum systems. Our approach is based on a charge-resolved master equation,
which is related to the Full-Counting statistics of charge detection. In the
quantum jump unravelling we show that this takes the form of a coupled system
of master equations, while for quantum diffusion it becomes a type of quantum
Fokker-Planck equation. In both cases, we show that the FPT can be obtained by
introducing absorbing boundary conditions, making their computation extremely
efficient. The versatility of our framework is demonstrated with two relevant
examples. First, we show how our method can be used to study the tightness of
recently proposed KURs for quantum jumps. Second, we study the homodyne
detection of a single two-level atom, and show how our approach can unveil
various non-trivial features in the FPT distribution.Comment: 8 pages, 2 figure