Does the transfer of energy take place when we push a huge rock with all our might and fail to move it?where the energy we spend is going?

Heat.

 

The rock won't move, and so in thermodynamic terms, nothing happens to it (unless you have warm hands, then it will locally heat up because of your hands).

 

But you spend a lot of energy by pushing. You muscles tense, and that means you will burn up sugars (forming CO2 and water, and heat). And your body will get rid of the heat, by breathing out hot air, and by sweating, or simply by being in contact with something cooler, like air.

 

So, eventually the energy we spend is going into the air, heating it up by a little bit.

Put another way, in thermodynamic terms, your engine did no work (the rock has no displacement) so your heat engine was 0% efficient. (Which means all of the energy was rejected as heat, as CaptainPanic has said.)