No, it is not possible for a quadratic equation to have no zeroes. It IS possible for one to have no real zeroes, and before you deal in complex numbers, that isgenerally what is meant by 'having no zeroes.' To determine if your zeroes are real or complex, use the determinant: d=b^2–4ac.
Since you are in 10,you will be learning about complex numbers which are the zeroes when determinant <0