We consider the null-controllability problem for the generalized Baouendi-Grushin equation (∂t − ∂ 2 x − q(x) 2 ∂ 2 y)f = 1ωu on a rectangular domain. Sharp controllability results already exist when the control domain ω is a vertical strip, or when q(x) = x. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general q and non-rectangular control region ω. In some geometries for ω, the upper bound and the lower-bound are equal, in which case, we know the exact value of the minimal time of null-controllability. Our proof relies on several tools: known results when ω is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the harmonic oscillator −∂ 2 x + ν 2 q(x) 2 when Re(ν) > 0, pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound.