What is a "conscious computer" ?
Consciousness is not a black and white line dividing entities in universe. This property can have varying levels. It emerges from a grade of potent design, which leads to intelligent behavior. The simplest feedback circuit can be regarded as close to the lowest level ( like simple micro-organisms). Much complex designs will achieve chaotic states and can be observed to be capable of near unpredictable, independent and intelligent looking behavior (like animals, humans).
As already stated in comments, self Replication is a possibility that comes along with the design. But it is not entirely possible on its own. For example:
Today's average computer can very well control a fully automated manufacturing facility to create its replicas, programmed to behave exactly same.
A more potently designed machine won't need a provided manufacturing facility, it will build it on its own from provided raw materials.
The next level will search for raw materials itself, provided how to identify them.
And so on.
But no matter how advanced the design be, there will always be something that has to be there already, or provided externally. Hence, conscious or not, designs can replicate, but still, no design is all-powerful, and no design is guaranteed to re-create itself exactly the same existence. for the "External element (pre-provided seed)" will always be unknown.