Can we feel the electrons which move from one object to other during rubbing which caused charge?

First of all , I don't understand what you mean by feeling electrons but here is an answer.

Can you feel the atoms in your finger with which you typed this question or the atoms in your brain where this question arised ?

No. The only way to know this happening or existing is through rigorous experiments and with the help of them we make a principal governing this. We know that electrons are exchanged when some substances are rubbed against other one because the substances show bizarre activities (all such attraction and repulsion).

Now coming to the second part.

I am unable to convince myself with the thought that why would electrons flow between two objects? What is the need?

There is no need. It was all you which caused the electron to come out of one atom and get absorbed by other atom. If you didn't rub them there will be no exchange of electrons.

One more thing I would like to tell you is that this exchange doesn't occur with any pair of substances and are related to a particular pairs only.

Talking about those particular pairs :

One of the bodies in those pairs need to be made up of atoms having less ionisation energy i.e. can loose electrons with a little supply of energy and the other one needs to be made up of atoms having appropriate or favourable electron affinity or greater tendency to absorb electrons .

So, the energy which you supply while rubbing is utilised by the atoms of one of the body and they release electrons and this electrons then jump to the atoms of the other body and become more stabilised and thus creating a net charge difference between the two surfaces.

So electrons come out because you supplied energy and were absorbed by other atoms since it was a more stable configuration (since electron gain enthalpy is greater than the ionisation energy).

Hope it helps ?.