Would the P vs. NP problem become trivial as a result of the development of universal quantum computers?
No, there will be absolutely no implication, for several reasons:


The P vs. NP problem is about classical computation rather than quantum computation. Even if quantum computers could solve NP-hard problems in polynomial time (which we don't expect them to be able to do), it could still be the case that classical computers cannot solve them in polynomial time.
Universal quantum computers, in a theoretical sense, are (to the best of my knowledge) already known to exist. These are just the quantum analogs of universal Turing machines: they can execute any given quantum "program".
Both quantum computation and the P vs. NP problem are theoretical notions. What someone can construct in the physical world has absolutely no bearing on anything having to do with them.


Lieuwe Vinkhuijzen gave a different interpretation of your question:


  Will quantum computers be able to solve NP-complete problems efficiently?


The expected answer is: no. So even in this sense, physical quantum computers won't enable us to solve NP-complete problems at will.