Tunable optical properties play an important role in the high performance of optoelectronic applications based on two-dimensional (2D) transition metal carbide and nitride (MXene) materials. Herein, the optical properties of functionalized MXene monolayers Sc2CT2 (T = O and OH) are investigated by strain engineering. The strain-dependent linear optical properties of Sc2CT2 possess broadband optical response due to the geometry and orbital overlap effect. The peaks from the second-order nonlinear coefficient elements d (d15, d16, and d31) at around half the band-gap exhibit a redshift for Sc2CO2 (blueshift for Sc2C(OH)2) with the increase of strain. The strain-dependent d reveals that Sc2CO2 with -1268 pm V-1 %-1 has a larger photoelastic coefficient than that of Sc2C(OH)2 with -574 pm V-1 %-1 at 1% strain. Meanwhile, the photoelastic tensors can not only be increased but also reduced with the increase of strain due to the dispersion relation. Moreover, the azimuthal angle-dependent second harmonic generation (SHG) from strained Sc2CT2 monolayers depends highly on the strained states and the pumping photon energy. The results pave the way for the tunable, broadband, and anisotropic applications of nonlinear optoelectronic devices based on MXenes based on strain engineering.