Were there any Elizabethan family feuds?

yes. Queen Elizabeth I and her sister Mary Queen of Scots faught between each other. this is why elizabeth was locked in a tower before she became queen.
there were also many others. a father and daughter would argue if they wouldn't agree on who the daughter should marry.  

Answer

Were there any Elizabethan family feuds?"  

Answer

Actually, Mary Queen of Scots was Mary Stuart. She was Elizabeth's cousin, whom Elizabeth very reluctantly had beheaded once she (Elizabeth) became queen, due to the threat Mary posed to Elizabeth's reign. (Mary Stuart and Elizabeth had a major religious conflict - Elizabeth was Protestant, while I believe Mary was Catholic - and therefore there were those who would depose Elizabeth and have Mary placed on England's throne, and Elizabeth could not allow that to happen.) Mary Tudor was Elizabeth's older half-sister. (I believe Mary was seventeen years Elizabeth's senior, but I mgiht be wrong.) Henry VIII (the girls' father) had declared his marriage to Catherine of Aragon (Mary's mother) void so that he could wed Anne Boleyn (Elizabeth's mother). He later beheaded Anne, I think primarily for adultery, although she was also charged with incest and witchcraft. For the brief period while Anne was queen, during Elizabeth's infancy, I seem to recall reading that Mary acted as Elizabeth's servant. While it might not seem like a strange thing to us that an older sibling should care for another one, with the nullification of his marriage to Catherine the king had also declared Mary's birth illegitimate. This meant that she lost the status of princess and only heir to the throne. While her birth was considered illegitimate - that is, while Henry said that he and Catherine had not been married at the time of Mary's birth - Mary could not become queen. Of course, throughout their lives, Henry declared and recanted several times that both Mary and Elizabeth were illegitimate. They fell in and out of his favor often.
For the record, Henry was abel to declare his marriage with Catherine null and void because he was the head of the Anglican Church (that is, the Church of England). In fact, he -founded- the Church when the Pope (England's previous religious authority) would not nullify the marriage. Of course, this was not the primary reason for the Anglican Church's founding; with Henry as head of both Church and State, England got twice the money she normally would from the people. Any funds that would have gone to the Pope in Rome now landed in England's coffers instead.
That's probably way more than you wanted to know. =P
-Blue Rusalka