The issues between Quakers and Puritans began long before the move to North America, with the same issues arising in the disputes both in old and New England.
Puritan ministers would actively go to Quaker sermons and meetings and try to start debates, and attack people for their views in the spaces that were made for those views to be expressed. They viewed Quakers as a disruption to their orthodoxy, and due to the extreme differences between the practices (with Puritans being tied closely to the church, and Quakers believing that the spirit is within the person and they do not need a church to worship) they saw Quakers as closer to witchcraft than a Christian religion.
Puritans believed that if they did not have control of their community then they were not fulfilling their religious calling, and they had to eliminate the beliefs in any opposing views.
Quakers were also relatively vocal and actively sought new members and their movement grew quite quickly. This caused public conflict and legal battles. George Fox, his wife, Margaret Fell, and William Penn, were all jailed and fined at various points throughout their life for preaching and practicing their religion. They were far from the only ones: in Puritan England, 3,000-4,000 Quakers were fined or imprisoned and 338 died in English prisons.