I think that the topic question should be broadened to "Can machines be made aware"? Computers are not brains in any sense of the word, although that it a common metaphor. The thing about computers is that they can be configured to perform the same kinds of calculations that human brains can. So, can they (or some future form of "computer") be used to build a genuinely sentient being? To answer that question, you need to analyze what it means for something to be sentient or self-aware.
Human minds are "embodied" in that they exist in a central nervous system that is connected to a peripheral nervous system. That is, brains have "sensors" and "actuators", to use robotic terminology. They become aware of their surroundings through sensor feeds (touch, smell, sound, taste, etc.), and they can move body parts through volition. Brains can also sense the position of body parts through
proprioception. Proprioception is a kind of self-awareness. In a sense, modern airliners and cars have a primitive type of self-awareness in that they contain sensors that constantly report on the condition, configuration, and location of their "body parts". Some robots can detect when their batteries are running low and that it is time to "feed" at a power outlet.
The field of robotics has been growing in leaps and bounds in recent years, thanks to advances in computer hardware and our understanding of how minds and brains function. We now know that walking robots need to think about where they plant their "feet", and they have to be able to report on failures of internal systems--just as humans need to be aware that pain signals a possible dangerous condition.
But robots are not self-aware in the sense that humans are, and they probably will not be for a very long time. Because people
tend to see themselves in everything, there is a strong tendency among those who do not know much about programming to think of computers as intelligent and possibly self-aware. We can build programs that learn about their surroundings, their present state, and possible future states. Robots do this all the time. So they have the deceptive appearance of thinking like humans do. I have seen robots navigate obstacle courses like they really seemed to be aware of their surroundings in the same way that we are. But they do not approach anything like our level of mental complexity in their calculations.
In order to achieve our level of self-awareness, a machine would need to be able not just to have awareness of its surroundings, but it would have to achieve "
metacognition"--awareness of its own thought processes. We really do not understand how that works in living brains yet, but there is no reason to think that there is anything non-physical or "quantum weird" in the process. We are making progress in unraveling the secrets of animal and human cognition.