[lum1n0l]

Will tetrasodium EDTA be converted into tetrammonium EDTA if it is combined with the stoichiometric equivalent amount of ammonia?

[Hunter2]

Not really or less.

You have in simple way
Na4ED + 4 NH3 + 4 H2O <=> ( NH4)4ED + 4 NaOH

The equilibrium is on the left side, you will still have  mostly the sodium salt, because NaOH is the stronger base. If you would start with the Ammonium salt, the NaOH would push out ammonia.
Ammonia is used to adjust pH for titration of Ca/Mg.

[Borek]

If done in water solution will contain all ions involved, so it will be impossible to say if it is Na₄EDTA with ammonia added, or (NH₄)₄EDTA with NaOH added (ignoring for a moment counterions and pH changes).

Whatever will drop out of the solution first on drying depends on individual salt solubilities.