Using the determinantal formula of Biane, Bougerol, and O'Connell, we give multi-time joint probability densities to the noncolliding Brownian motion with drill, where the number of particles is finite. We study a special case such that the initial positions of particles are equidistant with a period a and the values of drift coefficients are well-ordered with a scale sigma. We show that, at each time t > 0, the single-time probability density of particle system is exactly transformed to the biorthogonal Stieltjes-Wigert matrix model in the Chern-Simons theory introduced by Dolivet and Tierz. Here, one-parameter extensions (theta-extensions) of the Stieltjes-Wigert polynomials, which are themselves q-extensions of the Hermite polynomials, play an essential role. The two parameters a and sigma of the process combined with time t are mapped to the parameters q and theta of the biorthogonal polynomials. By the transformation of normalization factor of our probability density, the partition function of the Chern-Simons matrix model is readily obtained. We study the determinantal structure of the matrix model and prove that, at each time t > 0, the present noncolliding Brownian motion with drill is a determinantal point process, in the sense that any correlation function is given by a determinant governed by a single integral kernel called the correlation kernel. Using the obtained correlation kernel, we study time evolution of the noncolliding Brownian motion with drift. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4758795]