They are not related except that they are both friends of Friar Lawrence.
Friar Lawrence entrusts Friar John with the important letter to Romeo, instructing him to deliver it to Romeo in Mantua. However, Friar John is unable to get the letter to Romeo in time due to being quarantined.
The plan is that his brother Friar, Friar John, will take a letter to Romeo. But Friar John is prevented from giving him the message.
Friar John
Well, that is a complicated question, because no letter like that is delivered: There was a letter to Romeo talking about the false-death plan, but it was never delivered. Friar Lawrence wrote that letter and Friar John was supposed to deliver it, but was unable to due to a quarantine. Balthasar is the one who tells Romeo that Juliet has died (although she has not... this is the false-death that she will awake from), but he just tells him; he doesn't deliver a letter.
Friar John was quarantined.
Friar Lawrence sends a letter to Romeo, who is in Mantua, detailing his plan to fake the death of Juliet, and have Romeo bring her to Mantua where they can both live happily ever after. He gives the letter to Friar John, who is supposed to bring the letter to Romeo ASAP. Friar John never makes it to Romeo, because he was quarantined due to an outbreak of the Plague. Therefore, he brought the letter back to Friar Lawrence. This messed up Friar Lawrence' plan, as he had already set it in motion.
He sends Friar John as a messenger.
His messenger, Friar John, was quarantined.
================================================================================================================================================================ ANSWER: In Verona, Friar John tells Friar Lawrence that he was unable to deliver his letter to Romeo due to an outbreak of an infection. =================================================================================================================================================================
He sends a friend and fellow friar, Friar John, with a letter.
the priest of Verona
Friar Laurence asks Friar John to deliver a letter to Romeo explaining the plan to fake Juliet's death.