It would be nice if the following worked:

dist = TransformedDistribution[x1 + x2, {x1 \[Distributed] NormalDistribution[\[Mu], \[Sigma]], 
   x2 \[Distributed] NakagamiDistribution[m, \[CapitalOmega]]}, 
  Assumptions -> {m >= 1/2, \[CapitalOmega] > 0, \[Sigma] > 0}]
PDF[dist, z]

But at least for Mathematica 12.0, it doesn't. That means doing this the old-fashioned way by integrating over the joint distribution of the two random variables:

Integrate[(2 E^(-((m x1^2)/\[CapitalOmega])) x1^(-1 + 2 m) (m/\[CapitalOmega])^m)/Gamma[m]
  E^(-((z - x1 - \[Mu])^2/(2 \[Sigma]^2)))/(Sqrt[2 \[Pi]] \[Sigma]), {x1, 0, \[Infinity]}, 
  Assumptions -> {m >= 1/2, \[CapitalOmega] > 0, \[Sigma] > 0}]

The result for the pdf is