Yes but it will not increase beyond the maximum of 20
Telepathic has three distinct effects. The first one says:
Increase your Intelligence, Wisdom, or Charisma score by 1, to a maximum of 20.
So you cannot increase the stat you pick over 20, because the increase is to a maximum of 20. If you pick Charisma, and your Charisma is already at 20, this will do nothing. As you under normal circumstances cannot have all 3 stats at 20, it would be a better idea to pick one of the stats that is not yet at 20.
The third benefit is
You can cast the Detect Thoughts spell, requiring no spell slot or components, and you must finish a long rest before you can cast it this way again. Your spellcasting ability for the spell is the ability increased by this feat. If you have spell slots of 2nd level or higher, you can cast this spell with them.
If you picked Charisma, the stat to cast this (and which determines the save DC for the target) is Charisma. If your Chrisma was already at 20 when you picked the feat, it still is the "ability increased by this feat", but the increase is to a max of 20 (so effectively 0, not 1). If you however picked another stat, say Intelligence, then you use your Intelligence bonus for the spell's saving throw DC.
The spellcasting ability is only used for determining the DC to save against when casting the spell. The Intelligence check that the creature later can make to end Detect Thoughts has nothing to do with your spellcasting ability. It is not a save. It is always a contested Intelligence check, no matter wich ability you picked for the feat's increase:
the creature can use its action on its turn to make an Intelligence check contested by your Intelligence check if it succeeds, the spell ends.
If you coincidentally chose to increase Intelligence with the feat, you benefit from a possible bonus increase there for this check, but not because Intelligence is your spellcasting ability, just because it may have increased your Intelligence bonus. If you picked Charisma, it does not affect this contested check at all.