Henry VIII beheading Anne Boleyn is still one of English history's most infamous stories, so I can't imagine how much would be discussed within a few decades of it happening. Elizabeth was probably too young to remember the incident itself, but she definitely learned about it and obviously felt one way or another about it. Do we have any indication how she felt?
I'm writing this from memory as I'm at work right now and all my books are at home, so I apologize for the briefness (mods, if it is too short please feel free to delete.) While, to my recollection, she never commented PUBLICLY about her father executing her mother, she was close to her Boleyn relatives and showed the favour at her court once she was queen. She was also known to wear a small pendant with her mother's portrait inside.
Her aunt Mary Carey (nee Boleyn, Anne's sister) had two children from her marriage to William Carey--Henry and Catherine (many speculate one or both were actually Henry VIII children, but that's a whole different topic.) Henry was knighted and then made 1st Baron Hunsdon shortly after Elizabeth's accession, became a Knight of the Garter later on, and was offered the title Earl of Ormond (which was a Boleyn family title) on his deathbed (he declined.) Catherine married Sir Francis Knollys and had 10+ kids, and all except the last one lived to adulthood. Catherine became chief Lady of the Bedchamber to Elizabeth, an incredibly close and intimate position. Both Henry and Catherine served Elizabeth until their deaths, and both were buried in honour in Westminster Abbey and Elizabeth paid for their funerals.
The only member of the extended Boleyn family she fell out with was the eldest daughter of Catherine Knollys, Lettice. Lettice was described to look very similar to Elizabeth when she (Liz) was younger--red hair, vivacious personality, charming, etc. But she incurred Elizabeth's wrath when she married the current favourite Robert Dudley, 1st Earl of Leicester. Elizabeth banished them both from court. In time Leicester was allowed to return and was eventually back in Elizabeth's good grace, but not Lettice. Until Elizabeth's death in 1603 they met only once and put on a good show for the court, but no one bought it and nothing changed.
After Leicester's death Lettice's son from her first marriage became the Queen's favouite--Robert Deveraux, 2nd Earl of Essex, called "Sweet Robin" by Elizabeth. However, he thought being the Queen's favourite meant he could do what he pleased. Long story short, his anger led him to believe he could force a meeting with Elizabeth when he was on the outs with her, he was proclaimed a traitor, and was executed.
If you want to read further I recommend these books by Alison Weir: The Life of Elizabeth I; The Lady in the Tower: The Fall of Anne Boleyn; Mary Boleyn: The Mistress of Kings. By Nicola Tallis: Elizabeth's Rival: The Tumultuous Tale of Lettice Knollys, Countess of Leicester.
TL;DR: while Elizabeth appeared to "put the past behind her" once she became queen, she routinely favoured and rewarded members of her Boleyn family...unless they crossed her.