It has been reported on some of the salaf that they would greet each other when the month of Ramadan entered and there is no problem with this. For example they would say,
A blessed month
بارك الله لك في شهرك
May Allah bless you in your month.
Or that which is similar to this. The response of the one who receives this greeting is to reply with the like of that which they were greeted with. For example they would say
و لك بمثل هذا
And for you the likes of this also
و هو مبارك عليك
And it is a blessing upon you as well.
Or, you can respond with that which would (bring good) to the mind of the one who greeted you.