There is a generalized perception that the availability of suitable particle accelerators installed in hospitals, as neutron sources, may be crucial for the advancement of Boron Neutron Capture Therapy (BNCT). Progress on an ongoing project to develop a Tandem-ElectroStatic-Quadrupole (TESQ) accelerator for Accelerator-Based (AB)-BNCT is described here. The project goal is a machine capable of delivering 30 mA of 2.5 MeV protons to be used in conjunction with a neutron production target based on the ⁷Li(p,n)⁷Be reaction slightly beyond its resonance at 2.25 MeV. A folded tandem, with 1.25 MV terminal voltage, combined with an ESQ chain is being designed and constructed. A 30 mA proton beam of 2.5 MeV are the specifications needed to produce sufficiently intense and clean epithermal neutron beams, based on the ⁷Li(p,n)⁷Be reaction, to perform BNCT treatment for deep-seated tumors in less than an hour. The first design and construction of an ESQ module is discussed and its electrostatic fields are investigated theoretically and experimentally. Also new beam transport calculations through the accelerator are presented.