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Alternatively to MichaelE2's interesting answer you might use

• NMinimize

   J[q_?NumericQ, m_?NumericQ] := # . # &[{q - NIntegrate[ 1/Sqrt[2 Pi] Exp[-z^2/ 2] Tanh[ (Sqrt[q] z + 1/2 m)/(1/2) ]^2, {z, -Infinity, -10, 10, Infinity}, Method -> "GlobalAdaptive"], m - NIntegrate[ 1/Sqrt[2 Pi] Exp[-z^2/ 2] Tanh[ (Sqrt[q] z + 1/2 m)/(1/2)], {z, -Infinity, -10, 10, Infinity}, Method -> "GlobalAdaptive"]}] 

  NMinimize[ Re@J[q, m], {q, m}] // Quiet ({1.1934910^-17, {q -> 0.530368, m -> 6.3477810^-9}})

addendum

• or FixedPointList

  intqm =. intqm[q_?NumericQ, m_?NumericQ] := { NIntegrate[ 1/Sqrt[2 Pi] Exp[-z^2/ 2] Tanh[ (Sqrt[q] z + 1/2 m)/(1/2) ]^2, {z, -Infinity, -10, 10, Infinity}, Method -> "GlobalAdaptive"], NIntegrate[ 1/Sqrt[2 Pi] Exp[-z^2/ 2] Tanh[ (Sqrt[q] z + 1/2 m)/(1/2)], {z, -Infinity, -10, 10, Infinity}, Method -> "GlobalAdaptive"]} FixedPointList[Apply[intqm, #] &, {1, 1}, 15]

Both results agree well with MichaelE2's answer!