Quadratic nodal point (QNP) in two dimensions has so far been reported only in nonmagnetic materials and in the absence of spin-orbit coupling. Here, by first-principles calculations and symmetry analysis, we predict stable QNP near Fermi level in a two-dimensional kagome metal-organic framework material, Cr₃(HAB)₂, which features noncollinear antiferromagnetic ordering and sizable spin-orbit coupling. Effective k·p and lattice models are constructed to capture such magnetic QNPs. Besides QNP, we find Cr₃(HAB)₂ also hosts six magnetic linear nodal points protected by mirror as well as C_2zT symmetry. Properties associated to these nodal points, such as topological edge states and quantized optical absorbance, are discussed.