It is widely believed that there indeed exists a prime between $n(n+1)$ and $(n+1)(n+2)$ for every integer $n\geq 1$.  This appears to be beyond the scope of existing methods, however.  Even the "easier" problem of find a prime between $n^3$ and $(n+1)^3$ for all integers $n\geq 1$ appears to be beyond the scope of existing methods.  Dudek recently [proved][1] that there exists a prime between $n^3$ and $(n+1)^3$ for all $n\geq \exp\exp(33.217)$.  I suspect that the current computable threshold for your proposed problem would be much higher.


  [1]: https://arxiv.org/abs/1401.4233