This toolbox and attached document purport to describe in detail the mathematical framework for a computationally efficient computation of nonlinear Froude-Krylov forces (NLFK) in 6 degrees of freedom (DoFs) for axisymmetric floating objects. Additionally, the document also acts as a reference manual to a set of Matlab scripts, forming a demonstration toolbox to show the capabilities of the NLFK approach and provide an easy, operative, and ready-to-use implementation of the method. The toolbox is openly available at DOI: https://doi.org/10.5281/zenodo.3533854. This document and the toolbox are licensed with a Creative-Commons-By-Attribution-Share-Alike (CC-BY-SA) license. Note that this is the first version of the toolbox, so any feedback and potential corrections are welcome. Moreover, the user is highly invited to contact the author for any doubt, ideas or suggestions, deeper investigation, higher-complexity problems, and eventually collaboration. Finally, note that this toolbox is the precursor of an open source software, coded in a lower-level coding language than Matlab, hence much faster, which will be virtually shared by mid 2021. This work has received funding from the European Research Council under the Horizon 2020 Programme (H2020-MSCA-IF-2018)/ grant agreement no 832140.