I have been wondering to find the answer of some fundamental questions in quantum mechanics and the answer to the above question will help me to clear doubts of quantum world
Yes, there are operators in classical mechanics; one can formulate classical mechanics in terms of Hilbert space and operators, the only difference with quantum operators is that classical operators is that the former commute, see https://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics for more information.
Operators do exist in classical physics. The thing is, we don't use them in the same way.
In mechanics, for example, compute the eigenvalues and eigenvectors of an operator is important to study vibrational modes. You also can find examples of diagonalization of operators in electrodynamics and heat propagation.
Diagonalization of differential operators, Green Functions, and other elements of QFT lore already exist in classical physics.
The interpretation of the spectra of the operators in QM is different, often they are possible states, but the difference lives in the physics, not in the mathematical structure.