Computers are constructed with semiconductors.
I use and write atomistic simulators and semi-classical (finite-element analysis based) simulators to simulate modern FET devices every day as part of my research. I literally use quantum mechanics to write the algorithms behind the simulator (DFT, NEGF in my case). Moreover, if you look at the manual for Synopsys Sentaurus (which is the semi-classical simulator), they have hundreds of pages describing the derivation of their mobility models and structural models from basic quantum theory.
In summary, modern transistors are most definitely constructed taking quantum mechanics into account because without it you wouldn't have transistors functioning at 90nm much less at 14nm. You wouldn't take into account gate-leakage, tunneling, stress, bandstructures, and a host of other properties of modern transistors.
What? So much for purifying the discussion. I know a little about the history of electronics and do not remember any discussion of quantum mechanics by those building the first computers, lasers or transistors. These were rather practical inventions based on easily observable experimentation.
The field of quantum physics certainly allows us to better understand how and why these devices work, and to improve them, but it is not responsible for their existence.