Low-temperature elastic properties of the clathrate compound Ce_3Pd_<20>Si_6 were investigated using ultrasonic measurements. Pronounced softening of elastic constants (C_<11>-C_<12>)/2 and C_<44> with decreasing temperature indicates a precursor of electric quadrupole ordering due to a Γ_8 quartet ground sate with an orbital degeneracy for 8c site of a Ce ion. A minimum of (C_<11>-C_<12>)/2 at T_C=170mK indicates a transition from paramagnetic phase I to an ordered phase III. Applied magnetic fields induce an antiferro-quadrupole (AFQ) ordered phase II with highly anisotropic character dep...