Yes, but only if they choose any of the Elemental Disciplines (or other class feature) that allows them to cast a spell.
In order to meet the requirement for Elemental Adept you must be able to cast at least one spell. That means that you have to actually currently be able to cast it.
Some elemental disciplines allow you to cast spells.
Thus, if you choose an elemental discipline that lets you cast a spell, then that absolutely counts for the requirement to be able to cast a spell. If you choose only disciplines that don't give you the ability to cast a spell (and you have nothing else that gives you this ability) then you do not qualify for the feat.
To be clear, simply being a Way of the Four Elements monk is not enough, you must have an ability from something that allows you to currently cast a spell. Choosing one of the elemental disciplines that grants you a spell is a great way to do this.
You'll know the disciplines that allow you to cast a spell because they will say some variation on:
You can [...] cast [spell name].
Currently, your friend does not qualify for the any feat that requires spellcasting
You say that your friend currently only has two elemental disciplines: Elemental Attunement and Fists of Unbroken Air. However, neither of these grants the ability to cast any spell, so they do not qualify for the feat currently.
Elemental Adept is useless without a spell to cast anyways
Spells you cast ignore resistance to acid damage. In addition, when you roll damage for a spell you cast that deals acid damage, you can treat any 1 on a damage die as a 2.
The feat's only effect is to boost damage from spells only. The feat would have literally no benefit if you couldn't cast a spell (that does that type of damage). This is, in fact, the reason that having the ability to cast a spell is a prerequisite for this feat.