There is evidence from the text that Tybalt has written a letter to the Montague household. Benvolio mentions the letter in Act 2, scene 4:
Tybalt, the kinsman of old Capulet,
Hath sent a letter to his father's house.
'His father' is obviously a reference to Romeo's father, since Benvolio and Mercutio are discussing him (Romeo) at this point. The content of the letter is not quite clear, but Mercutio assumes (on his life) that it must be a challenge. Benvolio believes that Romeo would respond to such an invitation.
When Mercutio mentions that,
Any man that can write may answer a letter.
Benvolio answers that it goes further than that - that Romeo would stand up to Tybalt's challenge. Mercutio, however, expresses doubt that Romeo is capable:
... is he a man to
When Benvolio questions Mercutio as to Tybalt's ability, he practically overwhelms him with a diatribe in which he expresses both Tybalt's skill as a swordsman, his status and style, displaying contrasting sentiments towards him. Mercutio clearly both admires and despises Tybalt.
One may therefore assume that the letter Tybalt wrote was a challenge to Romeo, perhaps for attending the Capulet ball uninvited. Tybalt also probably felt hard done by when he was admonished by Capulet for wanting to pick a fight with Romeo when he saw him. He feels that his honour has been tainted and he therefore needs to defend it.
The most significant letter mentioned in the play is the one Friar Laurence had written to Romeo. This is mentioned in Act 4, scene 1, when the friar addresses Juliet in his cell.
In the mean time, against thou shalt awake,
Shall Romeo by my letters know our drift, ...
The idea was that Juliet, who was to agree to marry Paris, would drink a potion which would put her in a death-like slumber. Her parents, on believing that she had died, would incarcerate her in the family-vault. The letter would inform Romeo about this and he would secretly return to Verona where he and the friar would witness her awakening in the burial chamber.
Alas! Romeo never receives the letter since the messenger is delayed. He hears the wrong information and believes that his true love is dead. He buys a deadly potion from an apothecary, sees Juliet's lifeless body in the burial chamber, and drinks the potion, thus committing suicide.
There is no contact between Montague (he is never referred to as "Lord" in the text) and Tybalt throughout Romeo and Juliet. Tybalt does, however, send a letter to Romeo, which may be the answer you're looking for.
In his letter, Tybalt demands a duel from Romeo for showing up at the Capulet ball uninvited (essentially "crashing" it).