The aim of this article is to present a growth-fragmentation process naturally embedded in a Brownian excursion from boundary to apex in a cone of angle 2π/3. This growth-fragmentation process corresponds, via the so-called mating-of-trees encoding arXiv:1409.7055, to the quantum boundary length process associated with a branching SLE6 exploration of a γ=sqrt(8/3) quantum disc. However, our proof uses only Brownian motion techniques, and along the way we discover various properties of Brownian cone excursions and their connections with stable Lévy processes. Assuming the mating of trees encoding, our results imply several fundamental properties of the γ=sqrt(8/3)-quantum disc SLE6-exploration.