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
 (*{1.19349*10^-17, {q -> 0.530368, m -> 6.34778*10^-9}}*)

**addendum** 

or `FixedPointList`

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

 FixedPointList[Apply[intqm, #] &, {1, 1}, 15]
 (*{..., {0.530368, 0.0000164061}, {0.530368, 7.70484*10^-6}}*)

Both results agree well with MichaelE2's answer!